Transformations of Graphs

1. Translations

Introduction

A translation is a transformation that moves a graph from one position to another without changing its shape, size, or orientation.

Translations are one of the most important transformations in mathematics because they help us understand how changing an equation affects the position of its graph.

For example, consider the quadratic function:

y = x²

Its graph is a parabola with a vertex at (0, 0).

If we change the equation to:

y = x² + 3

the entire parabola moves 3 units upward.

If we change the equation to:

y = (x − 4)²

the parabola moves 4 units to the right.

These changes are examples of vertical and horizontal translations.

Translations are useful when studying linear functions, quadratic functions, absolute value functions, exponential functions, trigonometric functions, and many other mathematical relationships.

In this topic, we will explore how to describe translations, predict their effects on graphs, relate equations to movements, and sketch translated functions accurately.

What Is a Translation?

A translation is a transformation in which every point of a graph moves the same distance in the same direction.

During a translation:

• The shape of the graph remains unchanged.

• The size of the graph remains unchanged.

• The orientation remains unchanged.

• Every point moves by the same horizontal and vertical amounts.

• The domain and range may change depending on the translation.

For example, suppose a point A is located at (2, 3).

If the point moves 4 units right and 2 units up, its new coordinates are:

A′ = (2 + 4, 3 + 2)

A′ = (6, 5)

The same movement can be applied to every point on a graph.

Coordinate rule for translations

A translation of h units horizontally and k units vertically follows the rule:

(x, y) → (x + h, y + k)

where:

• h > 0 means moving right.

• h < 0 means moving left.

• k > 0 means moving up.

• k < 0 means moving down.

For example:

(x, y) → (x + 3, y − 2)

represents a translation 3 units right and 2 units down.

Visualising a Translation

Although this illustration uses a triangle, exactly the same principle applies to graphs of functions.

Horizontal Translations

A horizontal translation moves a graph to the left or right.

Suppose the original function is:

y = f(x)

A horizontal translation is represented by:

y = f(x − h)

where h is the horizontal movement.

• If h is positive, the graph moves right.

• If h is negative, the graph moves left.

This can initially seem confusing because the sign inside the brackets appears opposite to the direction of movement.

Horizontal translation rules

Why does the sign appear reversed?

Consider:

y = x²

The original vertex occurs when x = 0.

Now consider:

y = (x − 3)²

The squared expression is zero when:

x − 3 = 0

x = 3

Therefore, the vertex moves from (0, 0) to (3, 0).

The graph moves right, even though the equation contains subtraction.

Worked Example: Translating a Parabola Right

Original function:

y = x²

Translated function:

y = (x − 2)²

The graph moves 2 units to the right.

Compare corresponding points:

Notice that every x-coordinate increases by 2 while the y-coordinate remains unchanged.

The graph's shape does not change. Only its position changes.

Worked Example: Translating a Graph Left

Original function:

y = |x|

Translated function:

y = |x + 4|

The graph moves 4 units left.

The original vertex is:

(0, 0)

The translated vertex is:

(−4, 0)

For example, the original point (2, 2) becomes:

(−2, 2)

The original point (−2, 2) becomes:

(−6, 2)

Every point moves left by 4 units.

Vertical Translations

A vertical translation moves a graph upward or downward.

For an original function:

y = f(x)

A vertical translation is represented by:

y = f(x) + k

where k is the vertical movement.

• If k is positive, the graph moves upward.

• If k is negative, the graph moves downward.

Vertical translation rules

Unlike horizontal translations, the sign of the constant outside the function directly matches the direction of movement.

Worked Example: Translating a Parabola Up

Original function:

y = x²

Translated function:

y = x² + 3

Every point moves 3 units upward.

The x-coordinates remain unchanged.

The y-coordinates increase by 3.

The vertex moves from (0, 0) to (0, 3).

Worked Example: Translating a Graph Down

Original function:

y = √x

Translated function:

y = √x − 4

The graph moves 4 units downward.

The original starting point is:

(0, 0)

The translated starting point is:

(0, −4)

Other points translate as follows:

The graph retains its original shape.

Combining Horizontal and Vertical Translations

A graph can be translated horizontally and vertically at the same time.

The general form is:

y = f(x − h) + k

where:

• h represents the horizontal translation.

• k represents the vertical translation.

For the point transformation:

(x, y) → (x + h, y + k)

Worked Example: Translating a parabola

Consider:

y = (x − 3)² + 2

Compare this with:

y = x²

The graph moves:

• 3 units right.

• 2 units up.

The original vertex is (0, 0).

The new vertex is (3, 2).

Other corresponding points include:

Worked Example: Left and down

Describe the transformation:

y = (x + 5)² − 4

Compared with y = x²:

• The +5 inside the bracket means 5 units left.

• The −4 outside the bracket means 4 units down.

The vertex is:

(−5, −4)

The shape and width of the parabola remain unchanged.

Predicting Translated Graphs

We do not always need to plot many points to predict the position of a translated graph.

Instead, we can identify important features of the original graph and move them according to the translation rule.

Important features include:

• Vertices.

• Turning points.

• Endpoints.

• Intercepts.

• Asymptotes.

• Points of symmetry.

• Maximum and minimum values.

Worked Example: Translating a linear function

Original function:

y = 2x + 1

Translate the graph 3 units right and 4 units up.

Use:

g(x) = f(x − 3) + 4

Substitute:

g(x) = 2(x − 3) + 1 + 4

Expand:

g(x) = 2x − 6 + 5

g(x) = 2x − 1

The original line and translated line have the same gradient, 2.

The translated line is parallel to the original.

An important observation is that different combinations of horizontal and vertical translations can produce the same final line. For a non-horizontal straight line, the individual translations cannot always be determined uniquely from the original and final equations alone.

Worked Example: Translating an exponential function

Original function:

y = 2ˣ

Translate the graph 2 units right and 3 units down.

The translated equation is:

y = 2⁽ˣ ⁻ ²⁾ − 3

The original horizontal asymptote is:

y = 0

After translating down by 3 units, the new horizontal asymptote is:

y = −3

The original point (0, 1) becomes:

(2, −2)

The graph retains its exponential shape.

Translations of Different Function Families

The same translation rules apply to many kinds of functions.

The general translation rule remains:

g(x) = f(x − h) + k

The original function's shape is preserved.

Translations and the Domain and Range

Translations can change the domain and range of a function.

The domain is the set of permitted input values.

The range is the set of possible output values.

Worked Example: Square root function

Original:

y = √x

Domain:

x ≥ 0

Range:

y ≥ 0

Now translate the graph 4 units right and 2 units down:

y = √(x − 4) − 2

For the square root to be defined over the real numbers:

x − 4 ≥ 0

Therefore:

Domain: x ≥ 4

The minimum output occurs when x = 4:

y = √0 − 2 = −2

Therefore:

Range: y ≥ −2

The starting point moves from (0, 0) to (4, −2).

Worked Example: Quadratic function

Original:

y = x²

Range:

y ≥ 0

Translated:

y = (x + 3)² − 5

The vertex is (−3, −5).

Because the parabola still opens upward:

Range: y ≥ −5

The domain remains all real numbers.

General rule

If the original function has domain D and range R, a translation right by h and up by k gives:

• New domain: every value in D increased by h.

• New range: every value in R increased by k.

This is particularly useful when translating functions with restricted domains or ranges.

How Translations Affect Intercepts

Translations move every point of a graph, but the new x- and y-intercepts do not always come directly from translating the original intercepts.

An intercept is defined by its position on an axis.

After translation, a point that was originally on an axis may no longer be on that axis.

Worked Example

Original:

y = x²

Translated:

y = (x − 2)² − 4

The vertex is (2, −4).

To find the x-intercepts, set y = 0:

0 = (x − 2)² − 4

(x − 2)² = 4

x − 2 = ±2

Therefore:

x = 0 or x = 4

The x-intercepts are:

(0, 0) and (4, 0)

To find the y-intercept, set x = 0:

y = (0 − 2)² − 4

y = 0

The y-intercept is:

(0, 0)

This example shows that intercepts should be recalculated from the translated equation when needed.

Sketching Translated Graphs

A systematic method makes sketching translations much easier.

Step 1: Identify the parent function

For example:

y = x²

Recognise the familiar U-shaped parabola.

Step 2: Identify the translations

For:

y = (x − 2)² + 3

The graph moves 2 units right and 3 units up.

Step 3: Move the key points

The vertex moves:

(0, 0) → (2, 3)

Other points:

(−1, 1) → (1, 4)

(1, 1) → (3, 4)

(−2, 4) → (0, 7)

(2, 4) → (4, 7)

Step 4: Plot the points

Mark the new vertex and several corresponding points.

Step 5: Draw the graph

Connect the points using the same shape as the original function.

Step 6: Label important features

Include the equation, vertex, axes, and any important intercepts or asymptotes.

Describing a Translation from Two Equations

Sometimes we are given an original function and a transformed function and must describe the movement.

Worked Example

Original:

y = x²

Transformed:

y = (x + 6)² + 4

The graph moves:

6 units left and 4 units up.

Worked Example

Original:

y = |x|

Transformed:

y = |x − 3| − 7

The graph moves:

3 units right and 7 units down.

Worked Example

Original:

y = √x

Transformed:

y = √(x + 2) + 5

The graph moves:

2 units left and 5 units up.

Finding the Equation from a Translation

We can also work backwards.

Worked Example: From a description

The graph of y = x² is translated 5 units right and 2 units down.

Start with:

y = f(x)

Apply:

g(x) = f(x − 5) − 2

Therefore:

y = (x − 5)² − 2

Worked Example: From a new vertex

A parabola has the same shape and orientation as y = x², but its vertex is (−3, 4).

The equation is:

y = (x + 3)² + 4

Worked Example: Translating an absolute value graph

The graph of y = |x| is translated 4 units left and 1 unit down.

The equation is:

y = |x + 4| − 1

The vertex is (−4, −1).

Applications of Translations

Graph translations are useful because they allow mathematical models to be adjusted without rebuilding them from the beginning.

Physics: Vertical motion

Suppose the height of an object is modeled by:

h(t) = −5t² + 20t

where h is height in metres and t is time in seconds.

If the object is launched from a platform 15 m above the original reference level, its height becomes:

H(t) = −5t² + 20t + 15

The new graph is translated 15 units upward relative to the original height-time graph.

The parabolic shape remains unchanged.

This model assumes the same initial vertical velocity and gravitational acceleration, with air resistance neglected.

Business: Fixed operating costs

Suppose the variable production cost for n products is:

C = 8n

If a business also has a fixed cost of $200, the new cost function is:

C = 8n + 200

The graph moves upward by 200 units.

The gradient remains 8, meaning the cost still increases by $8 per additional product.

Biology: Shifting a growth model

Suppose an idealised growth model is:

P(t) = 100(1.1)ᵗ

If the same growth pattern begins 3 years later, its equation can be written:

P(t) = 100(1.1)⁽ᵗ ⁻ ³⁾

This represents a horizontal translation 3 years to the right.

The model has been shifted in time without changing its exponential growth pattern.

Comparing Translations with Other Transformations

Not every change to an equation represents a translation.

Consider the parent function:

y = x²

A translation changes position without changing shape or orientation.

A stretch or compression changes the dimensions of the graph.

A reflection changes its orientation.

These distinctions become especially important when studying combined function transformations.

Common Mistakes

Confusing horizontal directions

For horizontal translations, the sign inside the function appears opposite to the movement.

y = f(x − 3) moves right 3 units.

y = f(x + 3) moves left 3 units.

Confusing vertical directions

Vertical translations follow the sign outside the function.

y = f(x) + 4 moves up 4 units.

y = f(x) − 4 moves down 4 units.

Changing the shape of the graph

Translations do not stretch, compress, or reflect a graph.

Moving only the vertex

Every point on the graph must move by the same amount.

Incorrectly translating coordinates

A translation right 4 and down 2 changes (x, y) to (x + 4, y − 2), not (x − 4, y + 2).

Forgetting brackets

The equations y = (x − 3)² and y = x² − 3 represent different translations.

Assuming intercepts simply remain intercepts

A translated point that was originally an intercept may no longer lie on an axis. Recalculate intercepts when necessary.

Ignoring domain and range changes

Horizontal translations may change the domain, while vertical translations may change the range.

Key Terms

• Transformation: A change to the position, size, or orientation of a graph or shape.

• Translation: A transformation that moves every point the same distance in the same direction.

• Horizontal translation: A movement left or right.

• Vertical translation: A movement up or down.

• Parent function: A basic function used as a reference for transformations.

• Translated function: A function obtained by shifting a parent function.

• Coordinate rule: A rule describing how point coordinates change under a transformation.

• Vertex: The turning point of a parabola or the corner of an absolute value graph.

• Domain: The set of allowed input values.

• Range: The set of possible output values.

• Intercept: A point where a graph crosses or touches a coordinate axis.

• Asymptote: A line that a graph approaches in a specified limiting direction.

• Function notation: A way of expressing a function using symbols such as f(x).

• Invariant shape: A shape that remains unchanged under a transformation.

• Graph sketching: Representing the main features of a function on coordinate axes.

Key Takeaways

• A translation moves every point of a graph by the same amount.

• Translations preserve shape, size, and orientation.

• Horizontal translations are represented by y = f(x − h).

• Vertical translations are represented by y = f(x) + k.

• The combined translation rule is y = f(x − h) + k.

• Positive h moves a graph right; negative h moves it left.

• Positive k moves a graph up; negative k moves it down.

• A point (x, y) becomes (x + h, y + k).

• Key points such as vertices, endpoints, and asymptotes help us sketch translated graphs.

• The domain and range may change after a translation.

• Intercepts should be recalculated when necessary.

• Translation rules apply to quadratic, linear, absolute value, square root, exponential, trigonometric, and other functions.

Check Your Understanding

Recall and Understanding

1. Define a translation and explain which properties of a graph remain unchanged.

2. Describe the difference between horizontal and vertical translations.

3. Explain why y = (x − 4)² represents a translation to the right rather than to the left.

4. State the general equation for translating y = f(x) by h units horizontally and k units vertically.

Application

5. Describe the translation from y = x² to y = (x − 5)².

6. Describe the translation from y = |x| to y = |x + 3| − 4.

7. The point (2, 7) is translated 4 units left and 6 units down. Find its new coordinates.

8. Write the equation of y = x² after translating it 3 units right and 5 units up.

9. The graph of y = √x is translated 2 units left and 3 units down. Write its new equation and state its domain and range.

10. The graph of y = 2ˣ is translated 4 units right and 2 units up. Write the transformed equation and state its horizontal asymptote.

Analysis and Challenge

11. A parabola has the same shape and orientation as y = x², but its vertex is (−4, 6). Write its equation.

12. The graph of y = (x − 2)² + 3 is translated 5 units left and 4 units down. Find the resulting equation and vertex.

13. A function passes through the points (−2, 3), (0, 1), and (4, 5). Find the coordinates of these points after translating the graph 3 units right and 2 units down.

14. A quadratic function is given by y = (x − 3)² − 9. Determine its vertex, x-intercepts, and y-intercept.

15. A student claims that y = (x + 5)² − 2 is obtained from y = x² by moving 5 units right and 2 units down. Explain the error and give the correct translation.

Answers to Check Your Understanding

1. A translation moves every point of a graph by the same amount and in the same direction. Its shape, size, and orientation remain unchanged.

2. A horizontal translation moves a graph left or right and changes the input of the function. A vertical translation moves a graph up or down and changes the output.

3. The expression x − 4 equals zero when x = 4, so the vertex moves from x = 0 to x = 4. Therefore, the graph moves 4 units right.

4. The general equation is:

y = f(x − h) + k

5. The graph moves 5 units right.

6. The graph moves 3 units left and 4 units down.

7.

(2, 7) → (2 − 4, 7 − 6)

(−2, 1)

8.

y = (x − 3)² + 5

9.

New equation:

y = √(x + 2) − 3

Domain:

x ≥ −2

Range:

y ≥ −3

10.

New equation:

y = 2⁽ˣ ⁻ ⁴⁾ + 2

Horizontal asymptote:

y = 2

11.

Vertex = (−4, 6)

y = (x + 4)² + 6

12.

Original vertex:

(2, 3)

Translate left 5:

x = 2 − 5 = −3

Translate down 4:

y = 3 − 4 = −1

New vertex:

(−3, −1)

New equation:

y = (x + 3)² − 1

13.

Apply:

(x, y) → (x + 3, y − 2)

(−2, 3) → (1, 1)

(0, 1) → (3, −1)

(4, 5) → (7, 3)

14.

Given:

y = (x − 3)² − 9

Vertex:

(3, −9)

For x-intercepts, set y = 0:

(x − 3)² − 9 = 0

(x − 3)² = 9

x − 3 = ±3

x = 0 or x = 6

x-intercepts:

(0, 0) and (6, 0)

For the y-intercept, set x = 0:

y = (0 − 3)² − 9 = 0

y-intercept:

(0, 0)

15.

The student has confused the direction of the horizontal translation.

The expression x + 5 means the graph moves 5 units left, not right.

The −2 outside the squared expression means the graph moves 2 units down.

Correct translation: 5 units left and 2 units down.

The new vertex is (−5, −2).

Final Summary

Translations allow us to move graphs horizontally and vertically while preserving their shape and orientation.

By understanding how changes inside and outside a function affect its graph, we can predict transformations, sketch graphs accurately, and connect algebraic equations with geometric movements.

The most important relationship is:

y = f(x − h) + k

This represents a translation of h units horizontally and k units vertically.

Mastering translations provides an essential foundation for studying more advanced function transformations, graph analysis, and mathematical modeling.

2. Reflections

Introduction

A reflection is a transformation that produces a mirror image of a graph across a line.

In coordinate geometry, two of the most important reflections are:

• Reflection across the x-axis.

• Reflection across the y-axis.

Reflections change the orientation of a graph while preserving its shape and size.

For example, consider the quadratic function:

y = x²

Its graph is a parabola opening upward.

If we reflect it across the x-axis, the equation becomes:

y = −x²

The new parabola opens downward.

Now consider:

y = (x − 2)²

Reflecting this graph across the y-axis gives:

y = (x + 2)²

The vertex moves from (2, 0) to (−2, 0).

Reflections are important because they help us understand how negative signs affect functions, how graphs behave under transformations, and how algebraic equations are connected to geometric symmetry.

In this topic, we will explore how to reflect graphs, predict reflected equations, compare original and transformed functions, and recognise reflections from graphs.

What Is a Reflection?

A reflection is a transformation in which every point on a graph is moved to the opposite side of a fixed line, called the line of reflection.

Each point and its reflected image are the same perpendicular distance from the line of reflection.

For example, if a point lies 3 units above the x-axis, its reflection across the x-axis lies 3 units below it.

During a reflection:

• The shape remains unchanged.

• The size remains unchanged.

• Distances between points remain unchanged.

• The graph's orientation is reversed.

• Points on the line of reflection remain fixed.

A reflection is therefore a rigid transformation, also called an isometry.

Visualising Reflections

The same principle applies to graphs of mathematical functions.

Reflections Across the x-Axis

A reflection across the x-axis changes the sign of every y-coordinate.

The coordinate rule is:

(x, y) → (x, −y)

For example:

• (2, 3) becomes (2, −3).

• (−4, 5) becomes (−4, −5).

• (3, −2) becomes (3, 2).

• (0, 0) remains (0, 0).

The x-coordinate does not change.

Equation rule

If the original function is:

y = f(x)

its reflection across the x-axis is:

y = −f(x)

The negative sign is placed outside the entire function.

This means every output is multiplied by −1.

Worked Example: Reflecting a parabola

Original:

y = x²

Reflection across the x-axis:

y = −x²

The original graph opens upward.

The reflected graph opens downward.

Every y-value changes sign.

Reflecting More Complicated Functions Across the x-Axis

The reflection rule applies to the entire function, not just one term.

Worked Example: Linear function

Original:

y = 2x + 3

Reflection across the x-axis:

y = −(2x + 3)

Distribute the negative sign:

y = −2x − 3

The original line has gradient 2 and y-intercept 3.

The reflected line has gradient −2 and y-intercept −3.

Worked Example: Quadratic function

Original:

y = x² − 4x + 1

Reflection across the x-axis:

y = −(x² − 4x + 1)

Expand:

y = −x² + 4x − 1

Every term changes sign.

Worked Example: Absolute value function

Original:

y = |x − 3| + 2

Reflection across the x-axis:

y = −(|x − 3| + 2)

Simplify:

y = −|x − 3| − 2

The original graph has a minimum at (3, 2).

The reflected graph has a maximum at (3, −2).

Reflections Across the y-Axis

A reflection across the y-axis changes the sign of every x-coordinate.

The coordinate rule is:

(x, y) → (−x, y)

For example:

• (3, 4) becomes (−3, 4).

• (−2, 5) becomes (2, 5).

• (4, −3) becomes (−4, −3).

• (0, 2) remains (0, 2).

The y-coordinate does not change.

Equation rule

If the original function is:

y = f(x)

its reflection across the y-axis is:

y = f(−x)

We replace every occurrence of x with −x.

Worked Example: Reflecting a linear function

Original:

y = 3x + 2

Replace x with −x:

y = 3(−x) + 2

Simplify:

y = −3x + 2

The original line has gradient 3.

The reflected line has gradient −3.

The y-intercept remains 2 because points on the y-axis are unchanged.

Worked Example: Reflecting a parabola

Original:

y = (x − 4)²

Reflect across the y-axis:

y = (−x − 4)²

Factor out −1 inside the bracket:

y = [−(x + 4)]²

Since squaring removes the negative sign:

y = (x + 4)²

The original vertex is (4, 0).

The reflected vertex is (−4, 0).

Reflecting Polynomial Functions Across the y-Axis

When reflecting a polynomial across the y-axis, replace every x with −x and simplify.

Worked Example

Original:

y = x³ + 2x² − 5x + 4

Replace x with −x:

y = (−x)³ + 2(−x)² − 5(−x) + 4

Simplify each term:

(−x)³ = −x³

(−x)² = x²

−5(−x) = 5x

Therefore:

y = −x³ + 2x² + 5x + 4

Notice:

• Odd powers of x change sign.

• Even powers of x do not change sign.

• Constant terms remain unchanged.

This pattern can make reflections of polynomial functions much faster.

Comparing Reflections Across the x-Axis and y-Axis

Although both transformations create mirror images, they affect coordinates and equations differently.

The distinction is especially important when reflecting functions that are not symmetric.

Even and Odd Functions

Some graphs have natural symmetry.

This can cause a reflection to produce exactly the same graph.

Even functions

A function is even if:

f(−x) = f(x)

Its graph is symmetric about the y-axis.

Examples include:

• f(x) = x²

• f(x) = x⁴

• f(x) = |x|

• f(x) = cos x

For example:

f(x) = x²

f(−x) = (−x)² = x²

Therefore, reflecting y = x² across the y-axis produces the same graph.

Odd functions

A function is odd if:

f(−x) = −f(x)

Its graph is symmetric about the origin.

Examples include:

• f(x) = x

• f(x) = x³

• f(x) = sin x

For example:

f(x) = x³

f(−x) = (−x)³ = −x³

Therefore:

f(−x) = −f(x)

A reflection of an odd function across the y-axis produces the same graph as reflecting it across the x-axis.

Reflecting Absolute Value Functions

Absolute value graphs are particularly useful for understanding reflections because they have a distinctive V-shape.

Worked Example: Across the x-axis

Original:

y = |x|

Reflected:

y = −|x|

The original graph opens upward.

The reflected graph opens downward.

The vertex remains at (0, 0).

Worked Example: Across the y-axis

Original:

y = |x − 3|

Reflected:

y = |−x − 3|

Simplify:

y = |x + 3|

The vertex moves from (3, 0) to (−3, 0).

Worked Example: A translated absolute value function

Original:

y = |x − 2| + 4

Reflect across the x-axis:

y = −(|x − 2| + 4)

y = −|x − 2| − 4

Reflect the original across the y-axis instead:

y = |−x − 2| + 4

y = |x + 2| + 4

These transformations produce different graphs.

Reflecting Square Root Functions

Square root functions have restricted domains, making them useful for understanding how reflections affect domain and range.

Worked Example: Across the x-axis

Original:

y = √x

Domain:

x ≥ 0

Range:

y ≥ 0

Reflect across the x-axis:

y = −√x

Domain:

x ≥ 0

Range:

y ≤ 0

The graph is now below or on the x-axis.

Worked Example: Across the y-axis

Original:

y = √x

Reflect across the y-axis:

y = √(−x)

For the square root to be defined:

−x ≥ 0

Therefore:

x ≤ 0

Domain:

x ≤ 0

Range:

y ≥ 0

The graph now extends to the left of the y-axis.

Worked Example: Reflecting a translated square root function

Original:

y = √(x − 4) + 2

Its starting point is (4, 2).

Reflect across the y-axis:

y = √(−x − 4) + 2

The starting point becomes:

(−4, 2)

The reflected domain is:

x ≤ −4

The range remains:

y ≥ 2

Reflecting Exponential Functions

Exponential graphs can also be reflected.

Consider:

y = 2ˣ

Reflection across the x-axis

y = −2ˣ

This means y = −(2ˣ).

The graph lies below the x-axis.

Its horizontal asymptote remains y = 0.

Reflection across the y-axis

y = 2⁽⁻ˣ⁾

This can also be written as:

y = (1/2)ˣ

The original function increases as x increases.

The reflected function decreases as x increases.

Reflecting Graphs with Translations

Reflections can be combined with horizontal and vertical translations.

Consider:

y = −(x − 3)² + 4

Starting with:

y = x²

We can describe the transformations as:

1. Translate 3 units right.

2. Reflect across the x-axis.

3. Translate 4 units up.

The final vertex is:

(3, 4)

The parabola opens downward.

The transformations must be interpreted carefully because changing their order can sometimes change the final result.

Worked Example: Combining transformations

Start with:

y = |x|

Reflect across the x-axis:

y = −|x|

Translate 2 units left:

y = −|x + 2|

Translate 3 units down:

y = −|x + 2| − 3

The final vertex is (−2, −3).

The graph opens downward.

Identifying Reflections from Graphs

Sometimes we are given two graphs rather than equations.

To determine whether one is a reflection of the other, compare corresponding points and key features.

Reflection across the x-axis

Look for:

• Matching x-coordinates.

• Opposite y-coordinates.

• Equal distances above and below the x-axis.

• Reversed vertical orientation.

For example:

Original vertex: (2, 5)

Reflected vertex: (2, −5)

This is consistent with reflection across the x-axis.

Reflection across the y-axis

Look for:

• Opposite x-coordinates.

• Matching y-coordinates.

• Equal distances left and right of the y-axis.

• Reversed horizontal orientation.

For example:

Original vertex: (4, −2)

Reflected vertex: (−4, −2)

This is consistent with reflection across the y-axis.

Important limitation

A single matching pair of points does not prove that an entire graph has been reflected.

To confirm a reflection, the same coordinate rule must apply to every corresponding point.

Reflecting Graphs Step by Step

A reliable sketching method is:

Identify the original function.

Recognise its basic shape and important features.

Choose the line of reflection.

Determine whether the reflection is across the x-axis or y-axis.

Apply the coordinate rule.

Across the x-axis:

(x, y) → (x, −y)

Across the y-axis:

(x, y) → (−x, y)

Reflect important points.

These may include vertices, endpoints, intercepts, and turning points.

Sketch the reflected graph.

Preserve the original shape and size.

Check symmetry.

Corresponding points should be equally distant from the reflection axis.

Reflections and the Domain and Range

Reflections can affect the domain and range of functions.

Reflection across the x-axis

The x-values remain unchanged, so the domain remains the same.

The y-values change sign, so the range is reflected across zero.

Example:

Original:

y = x² + 2

Domain: all real numbers

Range: y ≥ 2

Reflected:

y = −x² − 2

Domain: all real numbers

Range: y ≤ −2

Reflection across the y-axis

The y-values remain unchanged, so the range remains the same.

The x-values change sign, so the domain is reflected across zero.

Example:

Original:

y = √(x − 3)

Domain: x ≥ 3

Range: y ≥ 0

Reflected:

y = √(−x − 3)

Domain: x ≤ −3

Range: y ≥ 0

These rules are particularly important for functions with restricted domains and ranges.

Reflecting Across Both Axes

A graph can be reflected across the x-axis and then across the y-axis.

Starting with:

y = f(x)

Reflect across the x-axis:

y = −f(x)

Then reflect across the y-axis:

y = −f(−x)

The coordinate rule is:

(x, y) → (−x, −y)

This is equivalent to a rotation of 180° about the origin.

Worked Example

Original:

y = x³ + 2

Reflect across both axes:

y = −f(−x)

Substitute:

y = −[(−x)³ + 2]

Simplify:

y = −[−x³ + 2]

y = x³ − 2

The final graph is equivalent to translating the original cubic graph downward by 4 units, although the transformation used here is a pair of reflections.

Applications of Reflections

Physics: Opposite directions

Suppose the vertical position of an object relative to a reference level is represented by:

y = f(t)

If we reverse the sign convention for vertical position, the new coordinate becomes:

y = −f(t)

The graph is reflected across the horizontal axis.

This is useful when comparing upward-positive and downward-positive coordinate systems.

Economics: Profit and loss

Suppose a function P(x) represents profit.

The function:

L(x) = −P(x)

represents the negative of profit, which may be interpreted as a loss balance under a suitable sign convention.

The graph of L(x) is the reflection of P(x) across the x-axis.

Signal processing

Signals can be reflected in two important ways.

Amplitude inversion:

g(t) = −f(t)

This reflects the signal across the horizontal axis.

Time reversal:

g(t) = f(−t)

This reflects the signal across the vertical axis.

These transformations are important in mathematics, physics, electronics, and signal analysis.

Common Mistakes

Confusing reflections across the two axes

Across the x-axis, change the sign of the output.

Across the y-axis, change the sign of the input.

Negating only one term

For an x-axis reflection:

y = −(x² + 3x + 2)

Correct:

y = −x² − 3x − 2

Not:

y = −x² + 3x + 2

Forgetting to replace every x

For a y-axis reflection, every occurrence of x must be replaced by −x.

Assuming all reflections change the graph

Some functions are symmetric about an axis, so reflecting them across that axis produces the same graph.

Confusing reflections with translations

Translations move graphs without reversing their orientation. Reflections produce mirror images.

Incorrectly simplifying powers

(−x)² = x²

(−x)³ = −x³

Ignoring domain and range changes

Reflections may change domain or range restrictions.

Assuming the reflection of every function is different

Even functions are unchanged by y-axis reflection.

Key Terms

• Reflection: A transformation that produces a mirror image across a line.

• Line of reflection: The fixed line across which a graph is reflected.

• x-axis reflection: A transformation changing (x, y) to (x, −y).

• y-axis reflection: A transformation changing (x, y) to (−x, y).

• Transformation: A change to the position, size, or orientation of a graph.

• Rigid transformation: A transformation preserving distances and shape.

• Coordinate rule: A rule describing how points move under a transformation.

• Function notation: A way of representing functions using symbols such as f(x).

• Even function: A function satisfying f(−x) = f(x).

• Odd function: A function satisfying f(−x) = −f(x).

• Symmetry: A property in which a graph matches itself under a specified transformation.

• Vertex: A turning point of a parabola or the corner of an absolute value graph.

• Domain: The set of permitted input values.

• Range: The set of possible output values.

• Intercept: A point where a graph meets a coordinate axis.

• Asymptote: A line that a graph approaches in a specified limiting direction.

• Amplitude inversion: A transformation changing f(x) to −f(x).

• Time reversal: A transformation changing f(t) to f(−t).

Key Takeaways

• Reflections produce mirror images of graphs.

• Reflecting across the x-axis changes (x, y) to (x, −y).

• Reflecting across the y-axis changes (x, y) to (−x, y).

• The equation for an x-axis reflection is y = −f(x).

• The equation for a y-axis reflection is y = f(−x).

• Reflections preserve the shape and size of a graph.

• Reflecting across the x-axis changes the sign of every output.

• Reflecting across the y-axis reverses the sign of every input.

• Even functions are symmetric about the y-axis.

• Odd functions are symmetric about the origin.

• Reflections can be combined with translations and other transformations.

• Domain and range restrictions must be considered when reflecting graphs.

• Reflected graphs can be identified by comparing corresponding points and their distances from the reflection axis.

Check Your Understanding

Recall and Understanding

1. Define a reflection and explain which properties of a graph remain unchanged.

2. State the coordinate rule for reflection across the x-axis.

3. State the coordinate rule for reflection across the y-axis.

4. Explain the difference between y = −f(x) and y = f(−x).

Application

5. Reflect the point (4, −7) across the x-axis.

6. Reflect the point (−3, 5) across the y-axis.

7. Write the equation obtained by reflecting y = 3x + 4 across the x-axis.

8. Write the equation obtained by reflecting y = x² − 6x + 2 across the y-axis.

9. Reflect y = |x − 5| + 3 across the x-axis. State the vertex of the reflected graph.

10. Reflect y = √(x − 4) across the y-axis. State the domain of the reflected function.

Analysis and Challenge

11. A parabola has equation y = (x − 3)² − 4. Reflect it across the y-axis and state its new vertex.

12. A function is defined by f(x) = x³ − 2x. Determine whether it is even, odd, or neither.

13. Reflect y = 2ˣ across both the x-axis and y-axis. Write the resulting equation.

14. A graph passes through (−2, 3), (0, −1), and (4, 5). Find the coordinates of the reflected points after reflection across the x-axis.

15. A student claims that reflecting y = x² + 4 across the y-axis produces y = −x² + 4. Explain the error and give the correct equation.

Answers to Check Your Understanding

1. A reflection produces a mirror image across a line. Shape, size, and distances between points remain unchanged.

2.

(x, y) → (x, −y)

3.

(x, y) → (−x, y)

4.

y = −f(x) reflects a graph across the x-axis by negating the outputs.

y = f(−x) reflects a graph across the y-axis by reversing the inputs.

5.

(4, −7) → (4, 7)

6.

(−3, 5) → (3, 5)

7.

Original:

y = 3x + 4

Reflected:

y = −(3x + 4)

y = −3x − 4

8.

Original:

y = x² − 6x + 2

Replace x with −x:

y = (−x)² − 6(−x) + 2

y = x² + 6x + 2

9.

Original:

y = |x − 5| + 3

Reflect across the x-axis:

y = −(|x − 5| + 3)

y = −|x − 5| − 3

Vertex:

(5, −3)

10.

Original:

y = √(x − 4)

Reflect across the y-axis:

y = √(−x − 4)

For the square root to be defined:

−x − 4 ≥ 0

Domain: x ≤ −4

11.

Original:

y = (x − 3)² − 4

Reflect across the y-axis:

y = (−x − 3)² − 4

Simplify:

y = (x + 3)² − 4

New vertex:

(−3, −4)

12.

f(x) = x³ − 2x

Calculate:

f(−x) = (−x)³ − 2(−x)

= −x³ + 2x

= −(x³ − 2x)

Therefore:

f(−x) = −f(x)

The function is odd.

13.

Original:

y = 2ˣ

Reflect across the x-axis:

y = −2ˣ

Reflect across the y-axis:

y = −2⁽⁻ˣ⁾

Equivalent form:

y = −(1/2)ˣ

14.

Reflect across the x-axis using:

(x, y) → (x, −y)

(−2, 3) → (−2, −3)

(0, −1) → (0, 1)

(4, 5) → (4, −5)

15.

The student has confused reflection across the y-axis with reflection across the x-axis.

For a y-axis reflection, replace x with −x:

y = (−x)² + 4

Since (−x)² = x²:

y = x² + 4

The original graph is symmetric about the y-axis, so the reflection produces the same graph.

Final Summary

Reflections are fundamental graph transformations that produce mirror images across coordinate axes.

Understanding how reflections affect coordinates and function equations allows us to predict, sketch, and compare transformed graphs accurately.

The two most important rules are:

Across the x-axis: y = −f(x)

Across the y-axis: y = f(−x)

These principles provide an important foundation for understanding symmetry, function transformations, and more advanced mathematical relationships.

3. Stretches and Compressions

Introduction

Not all graph transformations involve moving a graph to a different position or reflecting it across an axis. Some transformations change the graph's shape by stretching or compressing it.

A stretch makes a graph larger in a particular direction, while a compression makes it smaller in that direction.

For example, consider the quadratic function:

y = x²

Its graph is a parabola with a vertex at (0, 0).

Now consider:

y = 3x²

For every value of x, the output is three times the original output. This is called a vertical stretch by a factor of 3.

Alternatively:

y = 0.5x²

Every output is half its original value. This is a vertical compression by a factor of 0.5.

These transformations are important because they allow us to change the shape of a graph while preserving its general mathematical structure.

Stretches and compressions appear in quadratic functions, absolute value functions, exponential functions, trigonometric functions, and many mathematical models used in science and engineering.

What Are Stretches and Compressions?

A stretch or compression is a transformation that changes the distance of points from a particular axis.

There are two main types:

• Vertical stretches and compressions: Change the y-coordinates of points.

• Horizontal stretches and compressions: Change the x-coordinates of points.

In this topic, our main focus is on vertical transformations, although horizontal transformations will also be introduced for comparison.

Unlike translations and reflections, stretches and compressions generally change the shape or dimensions of a graph.

However, some points may remain unchanged, especially points lying on the relevant axis.

Vertical Stretches

A vertical stretch moves points farther from the x-axis.

For an original function:

y = f(x)

A vertical stretch by a factor of a is represented by:

y = af(x)

where:

a > 1

Every y-coordinate is multiplied by a.

The x-coordinates remain unchanged.

Coordinate rule

(x, y) → (x, ay)

For example, a vertical stretch by a factor of 3 transforms:

• (1, 2) → (1, 6)

• (2, 4) → (2, 12)

• (−1, −3) → (−1, −9)

• (0, 0) → (0, 0)

Points above the x-axis move farther upward.

Points below the x-axis move farther downward.

Points on the x-axis remain fixed.

Worked Example: Vertical Stretch of a Quadratic Function

Original function:

y = x²

Transformed function:

y = 2x²

The transformed function is a vertical stretch by a factor of 2.

Compare the values:

Every output is doubled.

The vertex remains at (0, 0).

The transformed parabola appears narrower than the original because its height increases more quickly as we move away from the vertex.

Vertical Compressions

A vertical compression moves points closer to the x-axis.

For an original function:

y = f(x)

a vertical compression is represented by:

y = af(x)

where:

0 < a < 1

Every y-coordinate is multiplied by a number between 0 and 1.

Coordinate rule

(x, y) → (x, ay)

For example, a vertical compression by a factor of 0.5 transforms:

• (1, 6) → (1, 3)

• (2, 8) → (2, 4)

• (−1, −4) → (−1, −2)

• (0, 0) → (0, 0)

Points move closer to the x-axis.

Worked Example: Vertical Compression of a Quadratic Function

Original:

y = x²

Transformed:

y = 0.5x²

The transformed function is a vertical compression by a factor of 0.5.

The outputs are halved.

The parabola appears wider because it increases more slowly as x moves away from zero.

Comparing Vertical Stretches and Compressions

The coefficient outside a function determines the vertical scaling.

The absolute value of the coefficient determines the scale.

The sign determines whether an additional reflection occurs.

General rule

For:

y = af(x)

• |a| > 1: vertical stretch.

• 0 < |a| < 1: vertical compression.

• a < 0: reflection across the x-axis as well.

• a = 1: no vertical scaling.

• a = −1: reflection across the x-axis only.

• a = 0: all outputs become zero, collapsing the graph onto the x-axis over its domain.

Stretches and Compressions of Absolute Value Functions

Absolute value functions provide a clear visual example of vertical scaling.

The parent function is:

y = |x|

Its graph has a V-shape with a vertex at (0, 0).

Worked Example: Vertical stretch

Consider:

y = 3|x|

Every output is multiplied by 3.

The graph becomes steeper, and its V-shape appears narrower.

Worked Example: Vertical compression

Consider:

y = 0.25|x|

Every output is multiplied by 0.25.

The graph becomes less steep and appears wider.

For example, the original point (4, 4) becomes (4, 1).

The vertex remains at (0, 0).

Worked Example: Scaling and reflection

Consider:

y = −2|x|

The coefficient −2 produces two transformations:

• A vertical stretch by a factor of 2.

• A reflection across the x-axis.

The graph opens downward and is steeper than y = |x|.

Stretches and Compressions of Linear Functions

Vertical scaling also applies to straight lines.

Consider:

f(x) = 2x + 1

Apply a vertical stretch by a factor of 3:

g(x) = 3f(x)

g(x) = 3(2x + 1)

g(x) = 6x + 3

The original gradient is 2.

The transformed gradient is 6.

The original y-intercept is 1.

The transformed y-intercept is 3.

Both the gradient and the y-intercept are multiplied by 3.

Worked Example: Vertical compression

Original:

y = 4x − 8

Compress vertically by a factor of 0.5:

y = 0.5(4x − 8)

y = 2x − 4

The gradient decreases from 4 to 2.

The y-intercept changes from −8 to −4.

An important observation is that a vertical stretch or compression of a straight line can also be described as a change in its gradient and intercept. Unlike a parabola, a straight line does not become visibly narrower or wider.

Horizontal Stretches and Compressions

Although our main focus is vertical scaling, it is useful to compare vertical transformations with horizontal ones.

For a function:

y = f(x)

a horizontal scaling can be written:

y = f(bx)

where b is a nonzero constant.

The horizontal scale factor is:

1/|b|

For positive b:

• b > 1 produces a horizontal compression.

• 0 < b < 1 produces a horizontal stretch.

For negative b, there is also a reflection across the y-axis.

Example: Horizontal compression

Original:

y = x²

Transformed:

y = (2x)²

Simplify:

y = 4x²

The horizontal scale factor is 1/2.

This is a horizontal compression by a factor of 1/2.

Interestingly, for this particular function, the same resulting graph can be described as a vertical stretch by a factor of 4.

Example: Horizontal stretch

Original:

y = x²

Transformed:

y = (x/2)²

Simplify:

y = 0.25x²

The horizontal scale factor is 2.

This is a horizontal stretch by a factor of 2.

It is also equivalent to a vertical compression by a factor of 0.25 for this particular function.

Comparing vertical and horizontal scaling

Remember: A coefficient outside the function scales outputs. A coefficient multiplying the input scales horizontal distances by its reciprocal.

Identifying Transformations from Equations

We can often identify stretches and compressions simply by examining the equation.

Worked Example: Quadratic function

Consider:

y = 5x²

Compared with y = x², the coefficient is 5.

Since 5 > 1, the graph undergoes a vertical stretch by a factor of 5.

Worked Example: Absolute value function

Consider:

y = 0.2|x|

Since 0 < 0.2 < 1, the graph undergoes a vertical compression by a factor of 0.2.

Worked Example: Negative coefficient

Consider:

y = −0.5x²

The coefficient has an absolute value of 0.5.

Therefore, the graph undergoes:

• A vertical compression by a factor of 0.5.

• A reflection across the x-axis.

The parabola opens downward and appears wider than y = x².

Combining Scaling with Translations

Stretches and compressions can be combined with translations.

A useful general form is:

y = af(x − h) + k

where:

• a controls vertical scaling and possibly reflection.

• h controls horizontal translation.

• k controls vertical translation.

Worked Example: Quadratic transformation

Consider:

y = 2(x − 3)² + 4

Compared with y = x²:

• Vertical stretch by factor 2.

• Translation 3 units right.

• Translation 4 units up.

The vertex is:

(3, 4)

The parabola opens upward and is narrower than y = x².

Worked Example: Compression and translation

Consider:

y = 0.5(x + 2)² − 3

Compared with y = x²:

• Vertical compression by factor 0.5.

• Translation 2 units left.

• Translation 3 units down.

The vertex is:

(−2, −3)

The parabola opens upward and is wider than y = x².

Worked Example: Stretch, reflection, and translation

Consider:

y = −3|x − 1| + 5

Compared with y = |x|:

• Vertical stretch by factor 3.

• Reflection across the x-axis.

• Translation 1 unit right.

• Translation 5 units up.

The vertex is:

(1, 5)

The graph is an inverted V-shape.

Sketching Stretched and Compressed Graphs

A reliable sketching method uses key points.

Step 1: Identify the parent function

For example:

y = x²

Step 2: Identify the scale factor

Suppose the transformed function is:

y = 3x²

The vertical scale factor is 3.

Step 3: Choose key points

For the parent function:

(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)

Step 4: Apply the coordinate rule

Multiply each y-coordinate by 3.

New points:

(−2, 12), (−1, 3), (0, 0), (1, 3), (2, 12)

Step 5: Plot the transformed points

Use the same coordinate scale for both graphs.

Step 6: Draw a smooth curve

The transformed graph should pass through the new points.

Step 7: Check the result

The new parabola should:

• Have the same vertex.

• Open upward.

• Be narrower than the original.

• Have outputs three times the original values.

Comparing Original and Transformed Graphs

When comparing graphs, examine their key features.

For quadratic functions, useful features include:

• Vertex.

• Axis of symmetry.

• Direction of opening.

• Width.

• Intercepts.

• Minimum or maximum value.

Worked Example

Compare:

y = x²

y = 4x²

y = 0.25x²

All three graphs:

• Have vertex (0, 0).

• Have axis of symmetry x = 0.

• Open upward.

• Have domain consisting of all real numbers.

• Have range y ≥ 0.

However:

• y = 4x² is narrower than y = x².

• y = 0.25x² is wider than y = x².

This demonstrates that vertical scaling can change a graph's appearance without changing its vertex or direction of opening.

Stretches and Compressions of Exponential Functions

Consider the parent function:

y = 2ˣ

Apply a vertical stretch by factor 3:

y = 3(2ˣ)

Every output is tripled.

For example:

At x = 0:

Original y = 1

Transformed y = 3

At x = 2:

Original y = 4

Transformed y = 12

The horizontal asymptote remains y = 0.

Vertical compression

Consider:

y = 0.5(2ˣ)

Every output is halved.

At x = 2:

Original y = 4

Transformed y = 2

The graph retains its exponential growth pattern.

For exponential functions, vertical scaling can sometimes also be expressed as a horizontal translation. For example:

2(2ˣ) = 2⁽ˣ ⁺ ¹⁾

The same graph can therefore have more than one valid transformation description.

Stretches and Compressions of Trigonometric Functions

Trigonometric functions provide an important application of vertical scaling.

Consider:

y = sin x

The maximum value is 1, and the minimum value is −1.

Its amplitude is 1.

Now consider:

y = 3 sin x

The maximum value becomes 3.

The minimum value becomes −3.

The amplitude is 3.

This is a vertical stretch by factor 3.

Vertical compression

Consider:

y = 0.5 sin x

The maximum value is 0.5.

The minimum value is −0.5.

The amplitude is 0.5.

This is a vertical compression by factor 0.5.

General trigonometric form

For:

y = a sin x

the amplitude is:

Amplitude = |a|

A negative value of a also reflects the graph across the x-axis.

For example:

y = −2 sin x

has amplitude 2 and is reflected across the x-axis compared with y = sin x.

Applications in Science and Engineering

Physics: Oscillations

A vibrating object may be modeled by:

y = A sin(ωt)

where:

• A = amplitude.

• ω = angular frequency.

• t = time.

Increasing A produces a vertical stretch.

Decreasing A produces a vertical compression.

For example:

y = 2 sin(3t)

has twice the amplitude of:

y = sin(3t)

The oscillation frequency remains unchanged.

Economics: Scaling revenue

Suppose a company's revenue is modeled by:

R(x) = 20x

If the selling price increases to $30 per unit, the new revenue function becomes:

R(x) = 30x

Since:

30x = 1.5(20x)

the new revenue graph is a vertical stretch of the original by a factor of 1.5.

This assumes the number of units sold remains the same at each value of x.

Biology: Comparing measurements

Suppose a laboratory instrument records a signal represented by:

S(t) = f(t)

If the signal is amplified threefold, the new signal is:

S(t) = 3f(t)

This is a vertical stretch by a factor of 3.

Signal amplification is common in scientific instruments, medical equipment, and electronic sensors.

Common Mistakes

Confusing stretches and compressions

A vertical stretch has a scale factor greater than 1.

A vertical compression has a scale factor between 0 and 1.

Confusing vertical and horizontal scaling

In y = 3f(x), the outputs are multiplied by 3.

In y = f(3x), horizontal distances are divided by 3.

Assuming a negative coefficient only produces scaling

A negative coefficient outside the function also produces reflection across the x-axis.

Multiplying x-coordinates during a vertical stretch

Vertical scaling changes y-coordinates, not x-coordinates.

Changing the vertex incorrectly

For y = ax², the vertex remains at (0, 0), regardless of the nonzero value of a.

Assuming every stretched graph becomes narrower

A vertical stretch makes a parabola such as y = x² appear narrower, but the visual effect depends on the type of function.

Ignoring brackets

The equations y = 2(x + 3)² and y = (2x + 3)² generally represent different transformations.

Using the wrong scale when sketching

Always use the same coordinate scale when comparing original and transformed graphs.

Key Terms

• Transformation: A change to a graph's position, size, or orientation.

• Vertical stretch: A transformation multiplying every output by a factor greater than 1.

• Vertical compression: A transformation multiplying every output by a positive factor less than 1.

• Horizontal stretch: A transformation increasing horizontal distances from the y-axis.

• Horizontal compression: A transformation decreasing horizontal distances from the y-axis.

• Scale factor: A number describing how distances or coordinates are multiplied.

• Parent function: A basic function used as a reference for transformations.

• Function notation: A way of writing functions using expressions such as f(x).

• Coordinate rule: A rule describing how points change during a transformation.

• Reflection: A transformation producing a mirror image across a line.

• Translation: A transformation moving every point the same distance and direction.

• Vertex: A turning point of a parabola or the corner of an absolute value graph.

• Axis of symmetry: A line dividing a graph into mirror-image halves.

• Gradient: The rate of change of a straight-line graph.

• Amplitude: The maximum displacement of an oscillating function from its midline.

• Domain: The set of allowed input values.

• Range: The set of possible output values.

• Asymptote: A line approached by a graph in a specified limiting direction.

Key Takeaways

• Vertical stretches and compressions change the y-coordinates of points.

• The general equation is y = af(x).

• If |a| > 1, the graph undergoes a vertical stretch.

• If 0 < |a| < 1, the graph undergoes a vertical compression.

• A negative coefficient also reflects the graph across the x-axis.

• Vertical scaling follows the coordinate rule (x, y) → (x, ay).

• Horizontal scaling is represented by y = f(bx), with scale factor 1/|b|.

• A vertical stretch makes y = x² appear narrower.

• A vertical compression makes y = x² appear wider.

• Stretches and compressions can be combined with translations and reflections.

• Key points and coordinate tables help us sketch transformed graphs accurately.

• Vertical scaling is important in modeling oscillations, financial relationships, and scientific signals.

Check Your Understanding

Recall and Understanding

1. Define a vertical stretch.

2. Define a vertical compression.

3. Explain the difference between y = 3f(x) and y = f(3x).

4. What additional transformation occurs when the coefficient outside a function is negative?

Application

5. Describe the transformation from y = x² to y = 5x².

6. Describe the transformation from y = x² to y = 0.2x².

7. The point (3, 4) undergoes a vertical stretch by a factor of 3. Find its new coordinates.

8. Write the equation obtained by vertically compressing y = |x| by a factor of 0.25.

9. Describe all transformations from y = x² to y = −2(x − 3)² + 4.

10. The graph of y = sin x undergoes a vertical stretch by a factor of 4. Write its equation and state its amplitude.

Analysis and Challenge

11. Compare y = x², y = 3x², and y = 0.5x². Which graph is narrowest, and which is widest?

12. A function passes through (−2, 3), (0, −4), and (5, 2). Find the coordinates after a vertical compression by a factor of 0.5.

13. A quadratic function is given by y = −0.5(x + 2)² + 3. Describe all transformations from y = x² and state the vertex.

14. The graph of y = 2x + 1 undergoes a vertical stretch by a factor of 3. Find the new equation, gradient, and y-intercept.

15. A student claims that y = f(4x) represents a horizontal stretch by a factor of 4. Explain the error and identify the correct transformation.

Answers to Check Your Understanding

1. A vertical stretch multiplies every y-coordinate by a factor greater than 1, moving nonzero outputs farther from the x-axis.

2. A vertical compression multiplies every y-coordinate by a positive factor less than 1, moving outputs closer to the x-axis.

3. y = 3f(x) is a vertical stretch by factor 3. y = f(3x) is a horizontal compression by factor 1/3.

4. A reflection across the x-axis.

5. Vertical stretch by factor 5.

6. Vertical compression by factor 0.2.

7.

(3, 4) → (3, 3 × 4)

(3, 12)

8.

y = 0.25|x|

9.

From y = x² to y = −2(x − 3)² + 4:

• Vertical stretch by factor 2.

• Reflection across the x-axis.

• Translation 3 units right.

• Translation 4 units up.

The vertex is (3, 4).

10.

Equation:

y = 4 sin x

Amplitude:

4

11.

Narrowest: y = 3x²

Widest: y = 0.5x²

The original y = x² lies between them in width.

12.

Apply:

(x, y) → (x, 0.5y)

(−2, 3) → (−2, 1.5)

(0, −4) → (0, −2)

(5, 2) → (5, 1)

13.

Given:

y = −0.5(x + 2)² + 3

Transformations:

• Vertical compression by factor 0.5.

• Reflection across the x-axis.

• Translation 2 units left.

• Translation 3 units up.

Vertex:

(−2, 3)

14.

Original:

y = 2x + 1

Stretch by factor 3:

y = 3(2x + 1)

y = 6x + 3

Gradient: 6

y-intercept: 3

15.

The student has confused the coefficient inside the function with the horizontal scale factor.

For y = f(4x), horizontal distances are multiplied by 1/4.

The correct transformation is:

Horizontal compression by a factor of 1/4.

Final Summary

Stretches and compressions change the dimensions of graphs by scaling their coordinates.

The most important vertical scaling relationship is:

y = af(x)

The magnitude of a determines whether the graph is stretched or compressed, while a negative sign introduces a reflection across the x-axis.

By understanding scale factors, comparing corresponding points, and recognising how coefficients affect functions, we can predict and sketch transformed graphs accurately.

These skills form an important foundation for studying combined transformations, function modeling, trigonometric amplitude, and more advanced graph analysis.

4. Combining Transformations

Introduction

Functions can undergo several transformations at the same time. A graph may be translated, reflected, stretched, or compressed, producing a new graph with a different position, orientation, or shape.

In earlier topics, we explored these transformations individually. However, many mathematical functions involve combinations of transformations.

For example, consider the parent function:

y = x²

Now consider:

y = −2(x − 3)² + 4

This function includes several transformations:

• A horizontal translation 3 units right.

• A vertical stretch by a factor of 2.

• A reflection across the x-axis.

• A vertical translation 4 units up.

The resulting parabola opens downward, is narrower than the original, and has a vertex at (3, 4).

Understanding combined transformations allows us to interpret complicated equations, sketch graphs efficiently, and explain how different functions are related.

These skills are important when studying quadratic, absolute value, square root, exponential, and trigonometric functions, as well as mathematical models used in science and engineering.

What Are Combined Transformations?

A combined transformation occurs when two or more transformations are applied to the same graph.

The main transformations include:

• Translations: Move a graph horizontally or vertically.

• Reflections: Produce a mirror image across an axis.

• Vertical stretches: Multiply outputs by a factor greater than 1.

• Vertical compressions: Multiply outputs by a positive factor less than 1.

• Horizontal stretches: Increase horizontal distances from the y-axis.

• Horizontal compressions: Decrease horizontal distances from the y-axis.

These transformations can be combined to produce a wide variety of functions.

The challenge is recognising which transformations are present and understanding how their order affects the final graph.

General Form of Combined Transformations

A useful general form is:

g(x) = af(b(x − h)) + k

where:

• f(x) is the parent function.

• a controls vertical stretching, compression, and possible reflection across the x-axis.

• b controls horizontal stretching, compression, and possible reflection across the y-axis.

• h represents horizontal translation.

• k represents vertical translation.

The effects of the parameters are:

This form provides a systematic way to describe multiple transformations.

Understanding the Order of Transformations

The order in which transformations are applied can affect the final graph.

For example, consider:

y = x²

Suppose we want to apply:

• A vertical stretch by factor 2.

• A vertical translation 3 units up.

Method A: Stretch first, then translate

Start with:

y = x²

Stretch vertically by factor 2:

y = 2x²

Translate upward by 3:

y = 2x² + 3

Method B: Translate first, then stretch

Start with:

y = x²

Translate upward by 3:

y = x² + 3

Stretch vertically by factor 2:

y = 2(x² + 3)

y = 2x² + 6

The two final equations are different.

Therefore:

The order of transformations matters.

Important principle

When interpreting a function written in the form:

g(x) = af(b(x − h)) + k

one convenient sequence is:

1. Apply the horizontal scaling and any y-axis reflection.

2. Apply the horizontal translation.

3. Apply the vertical scaling and any x-axis reflection.

4. Apply the vertical translation.

This sequence is especially useful when transforming coordinates.

Other sequences can produce the same final graph if the translation distances are adjusted appropriately.

Coordinate Rules for Combined Transformations

Instead of transforming the entire equation repeatedly, we can transform individual points.

For:

g(x) = af(b(x − h)) + k

the coordinate rule is:

(x, y) → (x/b + h, ay + k)

provided b ≠ 0.

This rule combines horizontal scaling, horizontal translation, vertical scaling, and vertical translation.

Worked Example

Consider the transformation:

g(x) = 2f(x − 3) + 4

Here:

a = 2

b = 1

h = 3

k = 4

The coordinate rule is:

(x, y) → (x + 3, 2y + 4)

Suppose the original graph passes through:

(1, 2)

The transformed coordinates are:

x′ = 1 + 3 = 4

y′ = 2(2) + 4 = 8

Therefore:

(1, 2) → (4, 8)

The same rule applies to every point on the original graph.

Worked Example: Transforming a Parabola

Start with:

y = x²

Transform it into:

y = −2(x − 3)² + 4

Step 1: Identify the transformations

The equation contains:

• A horizontal translation 3 units right.

• A vertical stretch by factor 2.

• A reflection across the x-axis.

• A vertical translation 4 units up.

Step 2: Identify the new vertex

The original vertex is:

(0, 0)

Apply the coordinate rule:

(x, y) → (x + 3, −2y + 4)

Therefore:

(0, 0) → (3, 4)

The new vertex is:

(3, 4)

Step 3: Transform additional points

Original points on y = x²:

(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)

Apply:

(x, y) → (x + 3, −2y + 4)

Step 4: Sketch the graph

Plot the transformed points and draw a smooth parabola.

The graph:

• Opens downward.

• Has vertex (3, 4).

• Has axis of symmetry x = 3.

• Is narrower than y = x².

• Has a maximum value of 4.

  

  Comparing Different Orders of Transformations

  Consider the parent function:

  y = |x|

  We will compare two transformation sequences.

  Sequence A: Reflect, then translate upward

  Reflect across the x-axis:

  y = −|x|

  Translate upward by 3:

  y = −|x| + 3

  The vertex is (0, 3).

  Sequence B: Translate upward, then reflect

  Translate upward by 3:

  y = |x| + 3

  Reflect across the x-axis:

  y = −(|x| + 3)

  y = −|x| − 3

  The vertex is (0, −3).

  The two graphs have different positions.

  This demonstrates that reflection and translation do not always commute.

  Two transformations commute when applying them in either order produces the same result.

  Some transformations commute, while others do not.

  For example, horizontal and vertical translations commute because they affect different coordinates.

  Combining Transformations of Absolute Value Functions

  Absolute value functions are useful for studying combined transformations because their V-shaped graphs have a clearly identifiable vertex.

  The parent function is:

  y = |x|

  Consider:

  y = −3|x + 2| + 5

  Step 1: Identify the transformations

  Compared with y = |x|:

  • Translate 2 units left.

  • Stretch vertically by a factor of 3.

  • Reflect across the x-axis.

  • Translate 5 units up.

  Step 2: Identify the vertex

  The parent vertex is (0, 0).

  The new vertex is:

  (−2, 5)

  Because the coefficient is negative, the graph opens downward.

  Step 3: Calculate key points

  Use:

  y = −3|x + 2| + 5

  x	y
  −4	−1
  −3	2
  −2	5
  −1	2
  0	−1

  

  Combining Transformations of Square Root Functions

  Consider the parent function:

  y = √x

  Its graph starts at (0, 0) and extends rightward and upward.

  Now consider:

  y = −2√(x − 4) + 3

  Identify the transformations

  • Translate 4 units right.

  • Stretch vertically by a factor of 2.

  • Reflect across the x-axis.

  • Translate 3 units up.

  Determine the starting point

  The original starting point is (0, 0).

  Using the coordinate rule:

  (x, y) → (x + 4, −2y + 3)

  The new starting point is:

  (4, 3)

  Transform other points

  Original point	Transformed point
  (0, 0)	(4, 3)
  (1, 1)	(5, 1)
  (4, 2)	(8, −1)
  (9, 3)	(13, −3)

  Domain and range

  For the square root to be defined:

  x − 4 ≥ 0

  Therefore:

  Domain: x ≥ 4

  Because the square root is non-negative and multiplied by −2:

  −2√(x − 4) ≤ 0

  Adding 3:

  Range: y ≤ 3

  The transformed graph starts at (4, 3) and curves downward to the right.

  Combining Horizontal and Vertical Scaling

  When a function contains coefficients both inside and outside its expression, we must distinguish horizontal and vertical scaling.

  Consider:

  g(x) = 2f(3x)

  The transformations are:

  • Horizontal compression by a factor of 1/3.

  • Vertical stretch by a factor of 2.

  The coordinate rule is:

  (x, y) → (x/3, 2y)

  For example, if the original function contains the point (6, 4), the transformed point is:

  (6/3, 2 × 4)

  (2, 8)

  Worked Example: Including translations

  Consider:

  g(x) = 3f(2(x − 4)) − 5

  Identify the parameters:

  a = 3

  b = 2

  h = 4

  k = −5

  Therefore:

  • Horizontal compression by a factor of 1/2.

  • Translation 4 units right.

  • Vertical stretch by a factor of 3.

  • Translation 5 units down.

  Coordinate rule:

  (x, y) → (x/2 + 4, 3y − 5)

  If the parent function contains the point (2, 3):

  x′ = 2/2 + 4 = 5

  y′ = 3(3) − 5 = 4

  The transformed point is:

  (5, 4)

  Why brackets matter

  Compare:

  g(x) = f(2(x − 4))

  and:

  g(x) = f(2x − 4)

  The first can be read as a horizontal compression by 1/2 followed by a translation 4 units right.

  The second can be rewritten as:

  g(x) = f(2(x − 2))

  It represents a horizontal compression by 1/2 followed by a translation 2 units right.

  The two equations do not describe the same transformation.

  Always factor the coefficient inside the function before identifying the horizontal translation.

  Combining Transformations of Exponential Functions

  Consider the parent function:

  y = 2ˣ

  Now consider:

  y = −3(2⁽ˣ ⁻ ²⁾) + 4

  Identify the transformations

  • Translate 2 units right.

  • Stretch vertically by a factor of 3.

  • Reflect across the x-axis.

  • Translate 4 units up.

  Transform key points

  The original function passes through:

  (0, 1), (1, 2), (2, 4)

  Apply:

  (x, y) → (x + 2, −3y + 4)

  Original point	Transformed point
  (0, 1)	(2, 1)
  (1, 2)	(3, −2)
  (2, 4)	(4, −8)

  Determine the asymptote

  The original horizontal asymptote is:

  y = 0

  The vertical stretch and reflection leave this asymptote at y = 0.

  The upward translation moves it to:

  y = 4

  The transformed graph approaches y = 4 from below.

  Its domain is all real numbers, and its range is:

  y < 4

  Combining Transformations of Trigonometric Functions

  Trigonometric functions often involve several transformations.

  Consider:

  y = 3 sin(2(x − π/4)) + 1

  Compared with y = sin x:

  • Vertical stretch by a factor of 3.

  • Horizontal compression by a factor of 1/2.

  • Translation π/4 units right.

  • Translation 1 unit up.

  The amplitude is:

  3

  The period is:

  2π/2 = π

  The midline is:

  y = 1

  The maximum value is:

  1 + 3 = 4

  The minimum value is:

  1 − 3 = −2

  7.7: Modeling with Trigonometric Functions - Mathematics LibreTexts

   

   

   

  These combined transformations are important in modeling sound waves, alternating electrical currents, vibrations, and periodic motion.

  Identifying Transformations from Equations

  A useful strategy is to compare the transformed equation with a familiar parent function.

  Example A

  y = 4(x + 3)² − 2

  Compared with y = x²:

  • Vertical stretch by factor 4.

  • Translation 3 units left.

  • Translation 2 units down.

  Example B

  y = −0.5|x − 6| + 1

  Compared with y = |x|:

  • Vertical compression by factor 0.5.

  • Reflection across the x-axis.

  • Translation 6 units right.

  • Translation 1 unit up.

  Example C

  y = √(2(x + 1)) − 4

  Compared with y = √x:

  • Horizontal compression by factor 1/2.

  • Translation 1 unit left.

  • Translation 4 units down.

  Example D

  y = −2f(−3(x − 5)) + 7

  Compared with y = f(x):

  • Horizontal compression by factor 1/3.

  • Reflection across the y-axis.

  • Translation 5 units right.

  • Vertical stretch by factor 2.

  • Reflection across the x-axis.

  • Translation 7 units up.

  The corresponding coordinate rule is:

  (x, y) → (−x/3 + 5, −2y + 7)

  Comparing Multiple Transformed Graphs

  Consider these three functions:

  A: y = (x − 2)² + 3

  B: y = 2(x − 2)² + 3

  C: y = −0.5(x − 2)² + 3

  All three have the same vertex:

  (2, 3)

  However, their shapes differ.

  Feature	Graph A	Graph B	Graph C
  Vertex	(2, 3)	(2, 3)	(2, 3)
  Opens	Upward	Upward	Downward
  Vertical scale	1	2	0.5
  Width	Standard	Narrower	Wider
  Extreme value	Minimum 3	Minimum 3	Maximum 3
  Axis of symmetry	x = 2	x = 2	x = 2

  This comparison shows how coefficients can change graph shape and orientation while leaving the vertex unchanged.

  Sketching Combined Transformations

  A reliable method for sketching transformed functions is to use key points from the parent graph.

  Step 1: Identify the parent function

  For example:

  y = x²

  Step 2: Identify the transformation parameters

  Suppose:

  y = −0.5(x + 2)² + 3

  Then:

  a = −0.5

  h = −2

  k = 3

  Step 3: Write the coordinate rule

  (x, y) → (x − 2, −0.5y + 3)

  Step 4: Select key points

  For y = x²:

  (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)

  Step 5: Transform the points

  Original	Transformed
  (−2, 4)	(−4, 1)
  (−1, 1)	(−3, 2.5)
  (0, 0)	(−2, 3)
  (1, 1)	(−1, 2.5)
  (2, 4)	(0, 1)

  Step 6: Draw and label the graph

  The graph is a downward-opening parabola with:

  • Vertex (−2, 3).

  • Axis of symmetry x = −2.

  • Maximum value 3.

  • A wider shape than y = x².

  Step 7: Verify the sketch

  Substitute a few plotted x-values into the equation to confirm their y-values.

  Determining an Equation from a Graph

  Sometimes we are given the final graph and must work backwards to identify its equation.

  Worked Example: Quadratic graph

  A parabola has:

  • Vertex (3, −2).

  • Opens downward.

  • Passes through (4, −5).

  Assume it has the form:

  y = a(x − h)² + k

  The vertex gives:

  h = 3

  k = −2

  Therefore:

  y = a(x − 3)² − 2

  Use the point (4, −5):

  −5 = a(4 − 3)² − 2

  −5 = a − 2

  a = −3

  The equation is:

  y = −3(x − 3)² − 2

  Compared with y = x², this graph has:

  • Vertical stretch by factor 3.

  • Reflection across the x-axis.

  • Translation 3 units right.

  • Translation 2 units down.

  Verification

  At x = 4:

  y = −3(1)² − 2

  y = −5 ✓

  Applications of Combined Transformations

  Physics: Projectile motion

  The height of a projectile may be modeled by:

  h(t) = −5(t − 2)² + 25

  This is a transformed quadratic function.

  Compared with y = x²:

  • Vertical stretch by factor 5.

  • Reflection across the x-axis.

  • Translation 2 units right.

  • Translation 25 units up.

  The vertex is:

  (2, 25)

  This means the projectile reaches a maximum height of 25 m after 2 seconds.

  The equation can also be expanded:

  h(t) = −5(t² − 4t + 4) + 25

  h(t) = −5t² + 20t + 5

  The initial height is 5 m.

  Curso Multimedia y web 2.0: MOVIMIENTO PARABOLICO

   

   

   

  Business: Revenue modeling

  Suppose a company's revenue is modeled by:

  R(x) = −2(x − 50)² + 5000

  This parabola opens downward.

  Its maximum occurs at:

  x = 50

  The maximum revenue is:

  R = 5000

  The vertex form makes it easy to identify the maximum without expanding the equation.

  The vertical stretch factor determines how rapidly revenue decreases as x moves away from the optimal value.

  Biology: Periodic measurements

  Suppose a periodic biological measurement is modeled by:

  P(t) = 5 sin(2(t − 1)) + 12

  This model includes:

  • Vertical stretch by factor 5.

  • Horizontal compression by factor 1/2.

  • Translation 1 unit right.

  • Translation 12 units up.

  The amplitude is 5.

  The midline is 12.

  The period is π time units.

  The model's maximum value is 17, and its minimum value is 7.

  This type of model can represent idealised rhythmic processes, although actual biological measurements may not follow perfect sine waves.

  Evaluating Transformation Descriptions

  A single transformed graph can sometimes be described using different combinations of transformations.

  For example:

  y = 4x²

  Compared with y = x², this is a vertical stretch by factor 4.

  But:

  y = (2x)² = 4x²

  Therefore, it can also be described as a horizontal compression by factor 1/2.

  Both descriptions are valid.

  Similarly, for exponential functions:

  2(2ˣ) = 2⁽ˣ ⁺ ¹⁾

  A vertical stretch by factor 2 produces the same graph as a translation 1 unit left.

  The equation and parent function must be considered together when identifying transformations.

  This is why different-looking transformation sequences can sometimes produce identical final graphs.

  Common Mistakes

  Applying transformations in the wrong order

  A vertical stretch followed by a vertical translation may produce a different result from performing the translation first.

  For example:

  2(x² + 3) = 2x² + 6

  is different from:

  2x² + 3

  Confusing horizontal translations

  In y = f(x − 4), the graph moves 4 units right, not left.

  Forgetting to factor horizontal coefficients

  For:

  y = f(3x − 6)

  rewrite the input as:

  3x − 6 = 3(x − 2)

  The horizontal translation is 2 units right, not 6 units right.

  Confusing vertical and horizontal scaling

  In y = 3f(x), outputs are multiplied by 3.

  In y = f(3x), horizontal distances are divided by 3.

  Forgetting reflections

  A negative coefficient outside a function introduces reflection across the x-axis.

  A negative coefficient multiplying the input introduces reflection across the y-axis.

  Moving the vertex without transforming other points

  Every point must follow the same coordinate transformation rule.

  Assuming all transformations change the graph visibly

  Some functions have symmetries that make particular reflections leave the graph unchanged.

  Ignoring domain and range restrictions

  A transformed square root function may have a new starting point and different domain or range.

  Confusing different but equivalent transformation descriptions

  For some function families, horizontal scaling can produce the same graph as vertical scaling.

  Failing to verify the final graph

  Check key points, intercepts, turning points, asymptotes, and domain restrictions.

  Key Terms

  • Combined transformation: Two or more transformations applied to a graph.

  • Parent function: A basic function used as the starting point for transformations.

  • Translation: A movement of a graph without changing its shape or orientation.

  • Reflection: A transformation producing a mirror image across a line.

  • Vertical stretch: Multiplying outputs by a factor greater than 1.

  • Vertical compression: Multiplying outputs by a positive factor less than 1.

  • Horizontal stretch: Increasing horizontal distances by a specified factor.

  • Horizontal compression: Decreasing horizontal distances by a specified factor.

  • Scale factor: A number describing how coordinates or distances are multiplied.

  • Coordinate transformation rule: A rule mapping original points to transformed points.

  • Vertex: A turning point of a parabola or the corner of an absolute value graph.

  • Axis of symmetry: A line dividing a graph into mirror-image halves.

  • Amplitude: The maximum displacement of a periodic function from its midline.

  • Period: The length of one complete cycle of a periodic function.

  • Midline: The central horizontal line around which a periodic function oscillates.

  • Domain: The set of permitted input values.

  • Range: The set of possible output values.

  • Asymptote: A line approached by a graph in a specified limiting direction.

  • Commutative transformations: Transformations that produce the same result when their order is reversed.

  • Equivalent transformations: Different transformation descriptions that produce the same final graph.

  Key Takeaways

  • Combined transformations involve applying several changes to the same function.

  • The general form g(x) = af(b(x − h)) + k describes vertical and horizontal scaling, reflections, and translations.

  • The coordinate rule is (x, y) → (x/b + h, ay + k), where b ≠ 0.

  • The order of transformations can affect the final graph.

  • Horizontal and vertical scaling must be distinguished carefully.

  • Factoring expressions inside a function helps identify horizontal translations correctly.

  • Negative scaling coefficients can introduce reflections.

  • Vertices, endpoints, intercepts, and asymptotes are useful for sketching transformed graphs.

  • Domain and range restrictions may change under transformations.

  • Comparing transformed graphs helps reveal the effects of individual parameters.

  • Different transformation sequences can sometimes produce the same final function.

  • Verifying key points and features helps ensure accurate graph sketches.

   

  Check Your Understanding

  Recall and Understanding

  1. What is meant by a combined transformation?

  2. Explain why the order of transformations can affect the final graph.

  3. State the coordinate transformation rule for g(x) = af(b(x − h)) + k.

  4. Explain the difference between a vertical stretch by factor 3 and a horizontal compression by factor 1/3.

  Application

  5. Describe all transformations from y = x² to:

  y = 3(x − 4)² + 2

  6. Describe all transformations from y = |x| to:

  y = −0.5|x + 3| − 2

  7. The point (2, 5) undergoes the transformation:

  (x, y) → (x + 4, −2y + 3)

  Find its new coordinates.

  8. Write the equation obtained by transforming y = x² using a vertical stretch by factor 2, a reflection across the x-axis, a translation 3 units left, and a translation 5 units up.

  9. Determine the vertex, axis of symmetry, and range of:

  y = −2(x − 1)² + 6

  10. Describe the transformations represented by:

  g(x) = 4f(2(x − 3)) − 1

  Analysis and Challenge

  11. A parabola has vertex (−2, 5), opens downward, and passes through (0, −3). Determine its equation in vertex form.

  12. A student claims that y = 2(x² + 4) and y = 2x² + 4 represent the same transformation. Explain why this is incorrect.

  13. Transform the points (−2, 4), (0, 0), and (2, 4) using:

  g(x) = −3f(2(x − 1)) + 5

  14. Compare the graphs:

  A: y = (x − 2)² + 1

  B: y = −2(x − 2)² + 1

  Describe their similarities and differences.

  15. The function y = √x is transformed into:

  y = −2√(x + 4) + 3

  Describe the transformations, state the starting point, and determine the domain and range.

  Answers to Check Your Understanding

  1. A combined transformation involves applying two or more transformations, such as translations, reflections, stretches, or compressions, to the same graph.

  2. Transformations can affect the positions of points differently depending on their order. For example, stretching a graph vertically after translating it upward also scales the translation distance, whereas stretching first and then translating does not.

  3.

  For:

  g(x) = af(b(x − h)) + k

  the coordinate rule is:

  (x, y) → (x/b + h, ay + k)

  where b ≠ 0.

  4. A vertical stretch by factor 3 multiplies y-coordinates by 3. A horizontal compression by factor 1/3 divides x-coordinates by 3. These are different transformations, although they can sometimes produce the same graph for particular functions.

  5.

  From y = x² to y = 3(x − 4)² + 2:

  • Vertical stretch by factor 3.

  • Translation 4 units right.

  • Translation 2 units up.

  Vertex: (4, 2).

  6.

  From y = |x| to y = −0.5|x + 3| − 2:

  • Vertical compression by factor 0.5.

  • Reflection across the x-axis.

  • Translation 3 units left.

  • Translation 2 units down.

  Vertex: (−3, −2).

  7.

  (x, y) → (x + 4, −2y + 3)

  For (2, 5):

  x′ = 2 + 4 = 6

  y′ = −2(5) + 3 = −7

  Answer: (6, −7)

  8.

  Start with y = x².

  Apply the stretch and reflection:

  y = −2x²

  Translate 3 units left:

  y = −2(x + 3)²

  Translate 5 units up:

  y = −2(x + 3)² + 5

  9.

  Given:

  y = −2(x − 1)² + 6

  Vertex:

  (1, 6)

  Axis of symmetry:

  x = 1

  The parabola opens downward, so its maximum value is 6.

  Range:

  y ≤ 6

  10.

  g(x) = 4f(2(x − 3)) − 1

  Transformations:

  • Horizontal compression by factor 1/2.

  • Translation 3 units right.

  • Vertical stretch by factor 4.

  • Translation 1 unit down.

  11.

  Use vertex form:

  y = a(x − h)² + k

  The vertex is (−2, 5), so:

  y = a(x + 2)² + 5

  Substitute (0, −3):

  −3 = a(0 + 2)² + 5

  −3 = 4a + 5

  4a = −8

  a = −2

  Therefore:

  y = −2(x + 2)² + 5

  Verification:

  At x = 0:

  y = −2(2²) + 5 = −3 ✓

  12.

  The equations are:

  y = 2(x² + 4)

  and:

  y = 2x² + 4

  Expand the first equation:

  y = 2x² + 8

  The second equation is:

  y = 2x² + 4

  They differ by 4 vertical units.

  In the first equation, the graph is translated upward by 4 and then stretched vertically by factor 2.

  In the second, the graph is stretched vertically by factor 2 and then translated upward by 4.

  The transformations do not produce the same graph.

  13.

  Given:

  g(x) = −3f(2(x − 1)) + 5

  Coordinate rule:

  (x, y) → (x/2 + 1, −3y + 5)

  For (−2, 4):

  x′ = −2/2 + 1 = 0

  y′ = −3(4) + 5 = −7

  Result: (0, −7)

  For (0, 0):

  x′ = 0/2 + 1 = 1

  y′ = −3(0) + 5 = 5

  Result: (1, 5)

  For (2, 4):

  x′ = 2/2 + 1 = 2

  y′ = −3(4) + 5 = −7

  Result: (2, −7)

  Transformed points: (0, −7), (1, 5), (2, −7)

  14.

  Graph A:

  y = (x − 2)² + 1

  Graph B:

  y = −2(x − 2)² + 1

  Similarities:

  • Both have vertex (2, 1).

  • Both have axis of symmetry x = 2.

  • Both have domain consisting of all real numbers.

  Differences:

  • Graph A opens upward.

  • Graph B opens downward.

  • Graph B is vertically stretched by factor 2 compared with Graph A.

  • Graph A has a minimum value of 1.

  • Graph B has a maximum value of 1.

  15.

  Original:

  y = √x

  Transformed:

  y = −2√(x + 4) + 3

  Transformations:

  • Translation 4 units left.

  • Vertical stretch by factor 2.

  • Reflection across the x-axis.

  • Translation 3 units up.

  Original starting point:

  (0, 0)

  New starting point:

  (−4, 3)

  Domain:

  x + 4 ≥ 0

  x ≥ −4

  Range:

  Because √(x + 4) ≥ 0:

  −2√(x + 4) + 3 ≤ 3

  y ≤ 3

   

  Final Summary

  Combining transformations allows us to construct and interpret complex functions using familiar mathematical operations.

  The general form:

  g(x) = af(b(x − h)) + k

  provides a framework for understanding horizontal and vertical scaling, reflections, and translations.

  By recognising the effects of each parameter, applying coordinate rules, and comparing key graph features, we can sketch transformed functions accurately and explain how they relate to their parent functions.

  The most important skill is understanding how the transformations work together, rather than memorising isolated rules.

5. Applications of Transformations

Introduction

Graph transformations are not simply techniques for moving or changing the shape of mathematical graphs. They are powerful tools for understanding and modeling real-world situations.

Many relationships in science, economics, engineering, biology, and technology can be represented using familiar mathematical functions. Instead of constructing a completely new model for every situation, we can often transform an existing function to describe new conditions.

For example, consider the quadratic function:

y = x²

This is a basic mathematical model with a U-shaped graph.

Now consider:

h(t) = −5(t − 2)² + 25

This transformed quadratic function could represent the height of a projectile over time.

The transformations tell us that:

• The graph opens downward because of the negative coefficient.

• The vertical scale is changed by a factor of 5.

• The maximum height occurs at t = 2 seconds.

• The maximum height is 25 metres.

By recognising the transformations, we can understand important features of the model without performing lengthy calculations.

In this topic, we will explore how transformations are used to solve problems, recognise patterns in real-world graphs, build mathematical models, and interpret the meaning of transformed functions.

Why Are Transformations Useful?

A transformation allows us to modify a familiar mathematical relationship to represent a different situation.

For example, suppose a business has a cost function:

C(x) = 12x

This represents a production cost of $12 per item.

If the business must also pay a fixed operating cost of $500, the new function becomes:

C(x) = 12x + 500

The graph is translated upward by 500 units.

We do not need to develop a completely new mathematical structure. We simply adjust the original function.

Transformations are useful because they allow us to:

• Adapt existing models to new conditions.

• Predict how graphs change when parameters change.

• Compare different scenarios.

• Identify maximum and minimum values.

• Interpret shifts in time or position.

• Understand changes in amplitude, scale, and growth.

• Simplify mathematical calculations.

• Make predictions using transformed functions.

Reviewing the Main Transformations

Consider a parent function:

y = f(x)

The main transformations are:

These transformations can be combined to represent more complicated situations.

General transformation model

A useful general equation is:

g(x) = af(b(x − h)) + k

where:

• a controls vertical scaling and possible reflection.

• b controls horizontal scaling and possible reflection.

• h controls horizontal translation.

• k controls vertical translation.

The values of these parameters can often be interpreted directly in a real-world context.

Recognising Transformations in Real-Life Graphs

When examining a real-world graph, look for changes in its important features.

Ask:

• Has the graph moved upward or downward?

• Does the same event occur earlier or later?

• Have the maximum and minimum values changed?

• Has the graph become steeper or flatter?

• Has the period of a repeating pattern changed?

• Has the graph been reflected?

• Has the general shape remained the same?

Example: Comparing two temperature models

Suppose the temperature in one location is modeled by:

T₁(t) = 8 sin(πt/12) + 25

A second location has the model:

T₂(t) = 8 sin(πt/12) + 29

The second function is:

T₂(t) = T₁(t) + 4

Therefore, the second temperature graph is translated vertically upward by 4°C.

The models have:

• The same amplitude.

• The same period.

• The same timing of maximum and minimum temperatures.

• Different average temperatures.

The second location is modeled as being 4°C warmer at every corresponding time.

This is a simple example of how a vertical translation can represent a real-world difference.

Application: Projectile Motion

Projectile motion is one of the clearest applications of transformed quadratic functions.

When an object moves vertically under approximately constant gravitational acceleration, its height can be modeled by:

h(t) = h₀ + v₀t − ½gt²

where:

• h(t) = height at time t.

• h₀ = initial height.

• v₀ = initial vertical velocity.

• g = gravitational field strength.

• t = time.

Near Earth's surface, g is approximately 9.8 m/s².

This equation is quadratic because it contains t².

Worked Example: A ball thrown upward

A ball is thrown upward from a platform 5 m above the ground with an initial vertical velocity of 20 m/s.

Using g = 10 m/s² for simplicity:

h(t) = 5 + 20t − 5t²

Rearrange:

h(t) = −5t² + 20t + 5

Step 1: Write in vertex form

Factor −5 from the quadratic terms:

h(t) = −5(t² − 4t) + 5

Complete the square:

t² − 4t = (t − 2)² − 4

Therefore:

h(t) = −5[(t − 2)² − 4] + 5

Expand:

h(t) = −5(t − 2)² + 25

Step 2: Interpret the transformations

Compared with y = x²:

• Reflect across the x-axis.

• Stretch vertically by a factor of 5.

• Translate 2 units right.

• Translate 25 units up.

Step 3: Identify the maximum height

The vertex is:

(2, 25)

Therefore:

Maximum height = 25 m

Time to maximum height = 2 s

Step 4: Find when the ball reaches the ground

Set h(t) = 0:

−5t² + 20t + 5 = 0

Divide by −5:

t² − 4t − 1 = 0

Apply the quadratic formula:

t = (4 ± √(16 + 4))/2

t = (4 ± √20)/2

t = 2 ± √5

The positive solution is:

t ≈ 4.24 s

The negative solution is not relevant because the model begins at t = 0.

The ball reaches the ground after approximately 4.24 seconds.

Why transformations are useful here

The vertex form makes the maximum height and its timing immediately visible.

We can also compare different launch conditions by examining how the parameters change.

For example, increasing the initial height while keeping the initial velocity unchanged translates the height-time graph upward.

Application: Business Revenue and Profit

Quadratic transformations are useful for modeling revenue and profit when these quantities depend on price or production.

Worked Example: Maximum profit

Suppose a business models its monthly profit using:

P(x) = −2(x − 40)² + 3200

where:

• x = number of products sold, measured in hundreds.

• P(x) = monthly profit in dollars.

Step 1: Identify the parent function

The parent function is:

y = x²

Step 2: Identify the transformations

The profit function has:

• A reflection across the x-axis.

• A vertical stretch by a factor of 2.

• A horizontal translation 40 units right.

• A vertical translation 3200 units up.

Step 3: Interpret the vertex

The vertex is:

(40, 3200)

Therefore:

Maximum modeled profit = $3,200

This occurs when x = 40, corresponding to 4,000 products sold.

Step 4: Compare different sales levels

At x = 30:

P(30) = −2(30 − 40)² + 3200

P(30) = −200 + 3200

P(30) = $3,000

At x = 50:

P(50) = −2(50 − 40)² + 3200

P(50) = $3,000

The model is symmetric about x = 40.

Interpretation

Selling 3,000 or 5,000 products gives the same modeled profit of $3,000.

The model suggests that profit decreases when sales move away from the optimal level.

This is an illustrative model; real business profit relationships may be more complicated.

Application: Comparing Production Costs

Linear functions are frequently used to model production costs.

Suppose a company has:

C₁(x) = 15x + 400

where:

• x = number of products.

• C₁(x) = total cost in dollars.

Now suppose the fixed operating cost increases by $200.

The new function becomes:

C₂(x) = 15x + 600

This is a vertical translation upward by 200 units.

Comparing the graphs

Both functions have gradient 15.

Therefore, the cost per additional product remains $15.

However, the second graph has a y-intercept of 600 instead of 400.

The difference between the functions is:

C₂(x) − C₁(x) = 200

This means the second production system costs $200 more at every production level.

Extension: Increasing variable costs

Suppose the variable cost increases from $15 to $18 per item, while the fixed cost stays at $400.

The new function is:

C₃(x) = 18x + 400

The gradient increases.

The graphs share the same y-intercept, but C₃(x) rises more quickly.

This is not a pure vertical stretch of the entire original function C₁(x), because a vertical stretch would also multiply the fixed cost.

Recognising this distinction is important when interpreting real-world models.

Application: Population Growth

Exponential functions are useful for modeling growth processes.

Suppose a population is modeled by:

P(t) = 500(1.08)ᵗ

where t is time in years.

The population begins at 500 and grows by 8% per year.

Scenario A: Doubling the initial population

If the initial population doubles but the growth rate remains unchanged:

P₂(t) = 1000(1.08)ᵗ

This can be written as:

P₂(t) = 2P(t)

The new graph is a vertical stretch by a factor of 2.

At every time t, the population is twice the original modeled population.

Scenario B: Growth begins 3 years later

Suppose the same growth pattern is delayed by 3 years.

A shifted model is:

P₃(t) = 500(1.08)⁽ᵗ ⁻ ³⁾

This is a horizontal translation 3 years to the right.

However, the model must be interpreted carefully.

If population growth is intended to begin only at t = 3, the shifted exponential model should be used for t ≥ 3, with the earlier period described separately.

Why transformations are useful

Transformations allow us to compare:

• Different starting populations.

• Different starting times.

• Different measurement scales.

• Growth patterns under different assumptions.

Application: Sound Waves and Oscillations

Many physical systems involve repeating patterns.

Examples include:

• Sound waves.

• Vibrating springs.

• Pendulums at small angles.

• Alternating electrical currents.

• Rotating machinery.

These patterns can often be modeled using sine and cosine functions.

The general form is:

y = A sin(B(t − C)) + D

where:

• A controls amplitude and possible reflection.

• B controls the period.

• C controls horizontal translation, or phase shift.

• D controls vertical translation.

The amplitude is:

|A|

The period is:

2π/|B|

for B ≠ 0.

Worked Example: A vibrating object

Suppose the displacement of a vibrating object is:

y = 4 sin(2πt) + 1

where y is measured in centimetres and t in seconds.

Step 1: Identify the amplitude

A = 4

Therefore:

Amplitude = 4 cm

Step 2: Identify the period

B = 2π

Period = 2π/(2π)

Period = 1 s

Step 3: Identify the vertical translation

D = 1

The graph is translated upward by 1 cm.

Its midline is:

y = 1

Step 4: Determine maximum and minimum displacement

Maximum:

1 + 4 = 5 cm

Minimum:

1 − 4 = −3 cm

Interpretation

The object oscillates around a position 1 cm above the original reference level.

Its maximum displacement from that midline is 4 cm.

It completes one full cycle every second.

Application: Electrical Signals

Electronic signals can be modified using transformations.

Suppose an electrical signal is represented by:

V(t) = 2 sin(100πt)

where V is voltage in volts and t is time in seconds.

Increasing the amplitude

If the signal is amplified by a factor of 3:

V₂(t) = 6 sin(100πt)

This is a vertical stretch by a factor of 3.

The amplitude increases from 2 V to 6 V.

Introducing a time delay

If the signal is delayed by 0.005 seconds:

V₃(t) = 2 sin[100π(t − 0.005)]

This is a horizontal translation 0.005 seconds to the right.

Reversing polarity

If the signal's polarity is reversed:

V₄(t) = −2 sin(100πt)

This is a reflection across the horizontal axis.

Why transformations are useful

Engineers use these mathematical ideas to understand:

• Signal amplification.

• Time delays.

• Phase differences.

• Signal inversion.

• Changes in frequency.

Application: Medical and Biological Signals

Some biological processes produce repeating electrical or mechanical patterns.

Examples include heartbeats, breathing cycles, and certain neural signals.

A simplified periodic signal might be modeled by:

S(t) = 2 sin(π(t − 1)) + 5

Compared with y = sin t:

• Vertical stretch by factor 2.

• Horizontal compression by factor 1/π.

• Translation 1 time unit right.

• Translation 5 units up.

The midline is 5.

The amplitude is 2.

The period is:

2π/π = 2 time units.

Real biological signals are often more complex than simple sine waves, but transformed functions can help describe basic features such as amplitude, timing, and frequency.

Application: Transformations in Geometry and Computer Graphics

Transformations are widely used in computer graphics, animation, engineering design, and digital mapping.

A point in a two-dimensional coordinate system can be transformed using coordinate rules.

Translation

Move a point h units horizontally and k units vertically:

(x, y) → (x + h, y + k)

Reflection across the x-axis

(x, y) → (x, −y)

Reflection across the y-axis

(x, y) → (−x, y)

Vertical scaling

(x, y) → (x, ay)

Horizontal scaling

(x, y) → (bx, y)

Worked Example: Transforming a digital shape

A triangle has vertices:

A(1, 2)

B(3, 2)

C(2, 5)

The triangle undergoes:

• A horizontal translation 4 units right.

• A vertical stretch by factor 2.

• A reflection across the x-axis.

Step 1: Write the coordinate rule

The combined rule is:

(x, y) → (x + 4, −2y)

Step 2: Transform each vertex

A(1, 2):

A′ = (5, −4)

B(3, 2):

B′ = (7, −4)

C(2, 5):

C′ = (6, −10)

Step 3: Interpret the result

The triangle has moved right, become twice as tall, and been reflected vertically.

Unlike a pure translation or reflection, the vertical stretch changes its shape.