1. Vertical Translations

Vertical Translations of Quadratic Functions

Introduction

A vertical translation is a transformation that moves a graph upward or downward without changing its shape or orientation.

For quadratic functions, vertical translations move the entire parabola while preserving its width, direction of opening, and axis of symmetry.

Consider the parent quadratic function:

y = x²

Its vertex is at (0, 0), and the parabola opens upward.

Now consider:

y = x² + 3

Every point on the original graph moves upward by 3 units.

Similarly:

y = x² − 4

Every point moves downward by 4 units.

Understanding vertical translations allows us to predict the position of a parabola, graph transformed functions efficiently, and compare quadratic relationships without recalculating every point.

The Parent Quadratic Function

The simplest quadratic function is:

f(x) = x²

It is called the parent function because other quadratic functions can be created by transforming it.

Characteristics

• Vertex: (0, 0)

• Axis of symmetry: x = 0

• Direction: Opens upward

• Minimum value: 0

• Domain: All real numbers

• Range: y ≥ 0

Table of Values

The graph is symmetric about the y-axis.

Understanding Vertical Translations

A vertical translation changes the y-coordinate of every point on a graph by the same amount.

The general rule is:

g(x) = f(x) + k

where k represents the vertical translation.

• If k > 0, the graph moves upward.

• If k < 0, the graph moves downward.

• If k = 0, the graph remains unchanged.

For the parent quadratic:

y = x² + k

Examples

Important Observation

The value of k is added to the output of the function, not the input.

This is why the transformation is vertical rather than horizontal.

Vertical Translation Upward

Consider:

y = x² + 3

The +3 means every y-value of the parent function increases by 3.

Comparing Values

Notice that every y-value increases by exactly 3.

Vertex

Original vertex: (0, 0)

Translated vertex: (0, 3)

What Remains Unchanged?

• The parabola opens upward.

• The width remains unchanged.

• The axis of symmetry remains x = 0.

• The domain remains all real numbers.

What Changes?

• The vertex moves upward.

• The minimum value increases from 0 to 3.

• The range becomes y ≥ 3.

• The graph no longer intersects the x-axis.

Vertical Translation Downward

Consider:

y = x² − 4

The −4 means every y-value decreases by 4.

Comparing Values

Vertex

Original vertex: (0, 0)

Translated vertex: (0, −4)

Important Changes

• The minimum value becomes −4.

• The range becomes y ≥ −4.

• The graph crosses the x-axis at x = −2 and x = 2.

Interpretation

The parabola has moved downward by four units, but its shape remains unchanged.

Comparing Vertical Translations

The following graph compares the parent parabola with upward and downward translations.

Key Observation

A vertical translation changes the position of the graph but not its shape.

The distance between corresponding points remains the same.

Predicting Changes to the Vertex

A parabola in vertex form is written as:

y = a(x − h)² + k

where:

• a determines the direction and vertical scale.

• h determines the horizontal position of the vertex.

• k determines the vertical position of the vertex.

The vertex is:

(h, k)

Example

Consider:

y = (x − 2)² + 5

The vertex is:

(2, 5)

The parabola has been shifted two units right and five units upward from y = x².

Another Example

Consider:

y = (x + 3)² − 4

Rewrite the bracket mentally as:

x − (−3)

Therefore:

h = −3

k = −4

The vertex is:

(−3, −4)

Important Observation

For vertical translations, only the k-value changes.

The h-value remains the same.

Translating a Parabola That Is Already Shifted

Vertical translations do not have to begin with the parent function.

Suppose:

f(x) = (x − 3)² + 2

The vertex is:

(3, 2)

Now translate the graph downward by five units.

Step 1: Identify the Existing k-Value

k = 2

Step 2: Subtract Five

2 − 5 = −3

Step 3: Write the New Function

g(x) = (x − 3)² − 3

New Vertex

(3, −3)

Interpretation

The parabola moves vertically from vertex (3, 2) to vertex (3, −3).

The horizontal position does not change.

Describing Vertical Translations Algebraically

Suppose the original function is:

f(x) = x² − 2x + 1

A vertical translation upward by four units is:

g(x) = f(x) + 4

Substitute:

g(x) = x² − 2x + 1 + 4

Therefore:

g(x) = x² − 2x + 5

Downward Translation

A downward translation by six units is:

g(x) = f(x) − 6

g(x) = x² − 2x + 1 − 6

g(x) = x² − 2x − 5

General Rule

If:

f(x) = ax² + bx + c

then a vertical translation by k units gives:

g(x) = ax² + bx + (c + k)

Only the constant term changes.

Understanding Why Only the Constant Changes

Consider:

f(x) = 2x² − 4x + 3

Translate upward by seven units.

g(x) = f(x) + 7

g(x) = 2x² − 4x + 3 + 7

g(x) = 2x² − 4x + 10

The coefficients of x² and x remain unchanged.

This means the parabola retains:

• The same shape.

• The same direction of opening.

• The same axis of symmetry.

The constant term increases because every output increases by seven.

Comparing Translated Graphs

Consider these functions:

f(x) = x²

g(x) = x² + 4

h(x) = x² − 2

Comparison

Observations

All three graphs have identical shapes.

The graph of g(x) is four units above f(x).

The graph of h(x) is two units below f(x).

Therefore, g(x) is six units above h(x).

Important Conclusion

Vertical translations preserve the shape and horizontal position of the parabola while changing its vertical position.

Vertical Translations and the Axis of Symmetry

The axis of symmetry passes vertically through the vertex.

For:

y = a(x − h)² + k

the axis of symmetry is:

x = h

Changing k does not change h.

Therefore, vertical translations do not change the axis of symmetry.

Example

Consider:

f(x) = (x − 4)² + 1

g(x) = (x − 4)² − 6

Both have:

Axis of symmetry: x = 4

Their vertices are:

f(x): (4, 1)

g(x): (4, −6)

The second graph is seven units below the first.

Vertical Translations and the Range

A vertical translation changes the range of a quadratic function.

Upward-Opening Parabolas

For:

y = a(x − h)² + k, where a > 0

the vertex is the minimum point.

The range is:

y ≥ k

Downward-Opening Parabolas

For:

y = a(x − h)² + k, where a < 0

the vertex is the maximum point.

The range is:

y ≤ k

Example

Consider:

y = −2(x + 1)² + 5

The parabola opens downward.

Its vertex is:

(−1, 5)

Its maximum value is 5.

Therefore:

Range: y ≤ 5

If translated downward by three units:

y = −2(x + 1)² + 2

The new range becomes:

y ≤ 2

Vertical Translations and x-Intercepts

Although a vertical translation does not change the shape of a parabola, it can change the number and position of its x-intercepts.

Example

Consider:

y = x² + 4

Set y = 0:

x² + 4 = 0

x² = −4

There are no real solutions.

Therefore, the graph has no x-intercepts.

Translate Downward

Now consider:

y = x² − 4

Set y = 0:

x² − 4 = 0

x² = 4

x = ±2

The graph now crosses the x-axis at:

(−2, 0) and (2, 0)

Important Observation

Vertical translations can create, remove, or change x-intercepts even though the shape of the parabola remains unchanged.

Vertical Translations and the y-Intercept

The y-intercept occurs where x = 0.

Consider:

f(x) = x² + 2x + 3

At x = 0:

f(0) = 3

The y-intercept is:

(0, 3)

Now translate the graph upward by five units:

g(x) = x² + 2x + 8

At x = 0:

g(0) = 8

The new y-intercept is:

(0, 8)

Conclusion

A vertical translation by k units changes the y-intercept by k units.

Recognising Vertical Translations from Equations

Sometimes we are given two quadratic functions and asked to describe the transformation.

Example

f(x) = 3x² − 2x + 1

g(x) = 3x² − 2x + 7

Compare the constant terms:

7 − 1 = 6

Therefore:

g(x) = f(x) + 6

The graph has been translated upward by six units.

Another Example

f(x) = −x² + 4x + 3

g(x) = −x² + 4x − 5

Compare:

−5 − 3 = −8

Therefore:

g(x) = f(x) − 8

The graph has been translated downward by eight units.

Distinguishing Vertical and Horizontal Translations

Vertical and horizontal translations affect equations differently.

Vertical Translation

g(x) = f(x) + k

The change occurs outside the function.

Example:

y = x² + 3

The graph moves upward three units.

Horizontal Translation

g(x) = f(x − h)

The change occurs inside the function.

Example:

y = (x − 3)²

The graph moves right three units.

Comparing the Transformations

Important Observation

Vertical translations change the output.

Horizontal translations change the input.

Real-World Application: Projectile Motion

Quadratic functions can model the height of an object moving through the air.

Example

Suppose the height of a ball is modelled by:

h(t) = −5t² + 20t

where:

• h = height above the ground in metres.

• t = time in seconds.

The ball is initially released at ground level.

Now suppose the same motion begins from a platform three metres above the ground.

The new height function is:

h(t) = −5t² + 20t + 3

Interpretation

The new graph is a vertical translation upward by three units.

At every time t, the ball is three metres higher than in the original model.

The shape of the trajectory remains unchanged.

Important Observation

This mathematical model assumes that the launch velocity and acceleration remain the same.

Only the initial height changes.

Real-World Application: Business Profit

Quadratic functions can be used to model profit.

Example

Suppose a company's profit is modelled by:

P(x) = −2x² + 40x − 100

where x represents the quantity produced.

Now suppose the company receives a fixed subsidy of $200.

The new profit function becomes:

Q(x) = −2x² + 40x + 100

Interpretation

The graph is translated upward by 200 units.

The quantity that maximises profit remains unchanged.

However, the maximum profit increases by $200.

Why?

Adding a constant amount to the profit at every production level changes the vertical position of the graph but not its shape.

Real-World Application: Engineering Design

Parabolic shapes are used in engineering and architecture.

Example

An arch is modelled by:

y = −0.5x² + 8

The vertex is:

(0, 8)

Suppose the entire arch is raised by two metres.

The new equation becomes:

y = −0.5x² + 10

Interpretation

The highest point increases from 8 metres to 10 metres.

The width and curvature of the arch remain unchanged.

The transformation is a vertical translation upward by two metres.

Common Mistakes

Confusing upward and downward shifts

A positive constant outside the function moves the graph upward.

A negative constant moves it downward.

Changing the x-coordinate of the vertex

A vertical translation changes only the y-coordinate.

Changing the shape of the parabola

Vertical translations do not stretch, compress, or reflect the graph.

Confusing vertical and horizontal translations

Changes outside the function are vertical.

Changes inside the function affect the input and may produce horizontal translations.

Forgetting to change the range

Vertical translations change the minimum or maximum value.

Assuming x-intercepts remain unchanged

Vertical translations may change the number and positions of x-intercepts.

Adding the translation to the wrong term

In standard form, a vertical translation changes the constant term.

Forgetting the original vertical position

When translating an already shifted parabola, add the new translation to its existing k-value.

Confusing the vertex with the y-intercept

These points are not always the same.

Assuming every parabola opens upward

The sign of the leading coefficient determines whether the parabola opens upward or downward.

Key Terms

• Vertical Translation: Movement of a graph upward or downward without changing its shape.

• Transformation: Change in the position, size, or orientation of a graph.

• Parent Function: Simplest function in a family of related functions.

• Quadratic Function: Function whose highest variable power is two.

• Parabola: Graph of a quadratic function.

• Vertex: Maximum or minimum point of a parabola.

• Axis of Symmetry: Vertical line dividing a parabola into matching halves.

• Vertex Form: Quadratic written as y = a(x − h)² + k.

• Vertical Shift (k): Amount a graph moves upward or downward.

• Horizontal Shift : Amount a graph moves left or right.

• Domain: Set of possible input values.

• Range: Set of possible output values.

• Minimum Value: Lowest value of a function.

• Maximum Value: Highest value of a function.

• x-Intercept: Point where a graph crosses or touches the x-axis.

• y-Intercept: Point where a graph crosses the y-axis.

• Translation Rule: Algebraic rule describing a graph's movement.

• Corresponding Points: Points occupying equivalent positions on an original and transformed graph.

• Function Notation: Representation such as f(x) or g(x).

• Quadratic Model: Quadratic function used to represent a real-world relationship.

Key Takeaways

• A vertical translation moves a parabola upward or downward.

• The general rule is g(x) = f(x) + k.

• Positive k-values produce upward translations.

• Negative k-values produce downward translations.

• Vertical translations do not change the shape of a parabola.

• The vertex moves vertically by the value of k.

• The x-coordinate of the vertex remains unchanged.

• The axis of symmetry remains unchanged.

• The domain of a quadratic function remains all real numbers.

• The range changes because the maximum or minimum value changes.

• Vertical translations can change the x-intercepts.

• In standard form, a vertical translation changes only the constant term.

• In vertex form, a vertical translation changes only k.

• Comparing equations can reveal the direction and distance of a translation.

• Vertical translations are useful in physics, engineering, business, and mathematical modelling.

Check Your Understanding

Identifying Vertical Translations

1. Describe the translation from y = x² to y = x² + 5.

2. Describe the translation from y = x² to y = x² − 7.

3. What happens to the vertex when a parabola moves upward four units?

4. Does a vertical translation change the shape of a parabola?

5. Does a vertical translation change the axis of symmetry?

Graphing and Vertices

6. Find the vertex of y = x² + 6.

7. Find the vertex of y = x² − 3.

8. Find the vertex of y = (x − 2)² + 5.

9. Find the vertex of y = (x + 4)² − 2.

10. Describe the translation from y = (x − 3)² + 2 to y = (x − 3)² − 4.

Algebraic Descriptions

11. Translate y = x² + 2 upward five units.

12. Translate y = x² − 3 downward four units.

13. Translate y = 2x² + 3x + 1 upward six units.

14. Translate y = −x² + 4x + 5 downward eight units.

15. Write the general rule for a vertical translation.

Comparing Graphs

16. Compare the vertices of y = x² + 2 and y = x² − 5.

17. Find the range of y = x² + 4.

18. Find the range of y = −x² + 3.

19. Explain why vertical translations can change x-intercepts.

20. Describe one real-world situation that can be modelled using a vertical translation.

Answer Key

Extended Challenge

Identifying Transformations

1. Describe the translation from y = x² + 3 to y = x² − 6.

2. Describe the translation from y = 2x² − 4 to y = 2x² + 7.

3. Determine the vertical translation from y = −3x² + 5 to y = −3x² − 2.

4. Explain why y = x² + 4 and y = (x + 4)² represent different transformations.

5. Describe what happens to the vertex and range when an upward-opening parabola is translated downward ten units.

Writing Equations

6. Translate y = x² − 2x + 3 upward seven units.

7. Translate y = 3x² + 4x − 5 downward six units.

8. Translate y = −2(x − 3)² + 4 upward nine units.

9. Translate y = (x + 5)² − 8 downward four units.

10. Write the equation of a parabola obtained by translating y = x² upward twelve units.

Comparing Graphs

11. Compare y = x² − 4 and y = x² + 6.

12. Determine the vertices and ranges of both functions.

13. Explain how their x-intercepts differ.

14. Sketch both parabolas on the same coordinate grid.

15. Explain why their axes of symmetry are identical.

Applications

16. A projectile follows h(t) = −5t² + 15t. Write a new equation if the projectile begins five metres above the original starting point.

17. A profit model is P(x) = −x² + 20x − 50. Write the new model after a fixed cost of $30 is removed.

18. A parabolic arch is described by y = −0.25x² + 12. Write its equation after being lowered three metres.

19. Explain why a vertical translation does not change the time at which a quadratic projectile model reaches its maximum height.

20. Describe how vertical translations can help compare related mathematical models.

Final Investigation: Comparing Vertical Translations

Consider the following quadratic functions:

Function A: y = x²

Function B: y = x² + 4

Function C: y = x² − 3

Function D: y = (x − 2)² + 5

Function E: y = (x − 2)² − 2

Complete the investigation.

1. Identify the parent quadratic function.

2. Describe the translation from A to B.

3. Describe the translation from A to C.

4. Determine the vertex of each function.

5. Determine the axis of symmetry of each function.

6. Identify which functions have the same axis of symmetry.

7. Create a table of values for Function A using x = −2, −1, 0, 1, 2.

8. Create a corresponding table for Function B.

9. Explain how the y-values of A and B are related.

10. Graph Functions A, B, and C on the same coordinate grid.

11. Compare their shapes.

12. Compare their ranges.

13. Determine the x-intercepts of Function C.

14. Explain why Function B has no real x-intercepts.

15. Describe the vertical translation from Function D to Function E.

16. Determine the difference between the vertices of D and E.

17. Graph Functions D and E on the same coordinate grid.

18. Explain why D and E have the same axis of symmetry.

19. Write a general rule describing vertical translations of quadratic functions.

20. Write a conclusion explaining how algebraic equations can be used to predict graphical transformations.

2. Horizontal Translations

Horizontal Translations of Quadratic Functions

Introduction

A horizontal translation is a transformation that moves a graph to the left or right without changing its shape, width, or direction of opening.

For quadratic functions, horizontal translations change the position of a parabola along the x-axis.

Consider the parent quadratic function:

y = x²

Its vertex is at (0, 0), and its axis of symmetry is x = 0.

Now consider:

y = (x − 3)²

This parabola has the same shape as the parent function but has moved three units to the right.

Similarly:

y = (x + 4)²

This parabola has moved four units to the left.

An important feature of horizontal translations is that the direction of movement may initially seem opposite to the sign inside the brackets.

Understanding this relationship allows us to graph translated parabolas, predict new vertex coordinates, describe transformations algebraically, and compare quadratic functions.

The Parent Quadratic Function

The parent quadratic function is:

f(x) = x²

Its graph is a U-shaped parabola. 

Characteristics

• Vertex: (0, 0)

• Axis of symmetry: x = 0

• Direction of opening: Upward

• Minimum value: 0

• Domain: All real numbers

• Range: y ≥ 0

Table of Values

The graph is symmetric about the y-axis.

Every point on the graph can be moved horizontally to create a translated parabola.

Understanding Horizontal Translations

A horizontal translation changes the x-coordinate of every point on a graph by the same amount.

The general transformation rule is:

g(x) = f(x − h)

where h represents the horizontal translation.

• If h > 0, the graph moves right.

• If h < 0, the graph moves left.

• If h = 0, the graph remains unchanged.

For the parent quadratic function:

y = (x − h)²

Examples

Important Observation

A subtraction inside the brackets moves the graph to the right.

An addition inside the brackets moves the graph to the left.

This is the opposite of what students may initially expect.

Why Does the Direction Seem Reversed?

Consider:

y = (x − 3)²

The vertex occurs where the squared expression equals zero.

Set:

x − 3 = 0

Therefore:

x = 3

The vertex is at (3, 0), so the parabola has moved three units to the right.

Now consider:

y = (x + 4)²

Set:

x + 4 = 0

Therefore:

x = −4

The vertex is at (−4, 0), so the parabola has moved four units to the left.

A Useful Rule

Think about the value of x that makes the expression inside the brackets equal to zero.

That value gives the horizontal coordinate of the vertex when the function is in vertex form.

Horizontal Translation to the Right

Consider:

y = (x − 3)²

This is the parent function translated three units to the right.

Table of Values

Vertex

Original vertex: (0, 0)

New vertex: (3, 0)

Axis of Symmetry

x = 3

What Remains Unchanged?

• Shape and width.

• Direction of opening.

• Minimum value.

• Range.

• Domain.

What Changes?

• Horizontal position.

• Vertex x-coordinate.

• Axis of symmetry.

• y-intercept.

Horizontal Translation to the Left

Consider:

y = (x + 2)²

This is the parent function translated two units to the left.

Table of Values

Vertex

Original vertex: (0, 0)

New vertex: (−2, 0)

Axis of Symmetry

x = −2

Interpretation

Every point on the original parabola moves two units left.

The y-values remain unchanged at corresponding translated points.

Exploring Horizontal Translations

Use the slider to compare the parent parabola with its horizontally translated graph.

Key Observation

The vertex moves horizontally, but its y-coordinate remains unchanged.

The entire parabola moves with the vertex.

Determining New Vertex Coordinates

The vertex form of a quadratic function is:

y = a(x − h)² + k

where:

• a controls the direction of opening and vertical scale.

• h determines the horizontal position.

• k determines the vertical position.

The vertex is:

(h, k)

Example

Consider:

y = (x − 4)² + 2

The vertex is:

(4, 2)

Now translate the graph three units to the right.

The new vertex becomes:

(7, 2)

The equation becomes:

y = (x − 7)² + 2

Another Example

Consider:

y = −2(x + 1)² − 5

The vertex is:

(−1, −5)

Translate the graph four units to the left.

New vertex:

(−5, −5)

New equation:

y = −2(x + 5)² − 5

Important Observation

Horizontal translations change h but leave k unchanged.

Describing Transformations Using Function Notation

Function notation provides a general way to describe horizontal translations.

Suppose:

f(x) = x² + 2x + 1

A translation three units to the right is written:

g(x) = f(x − 3)

Step 1: Replace x with (x − 3)

g(x) = (x − 3)² + 2(x − 3) + 1

Step 2: Expand

g(x) = x² − 6x + 9 + 2x − 6 + 1

Step 3: Simplify

g(x) = x² − 4x + 4

Interpretation

The graph of g(x) is the graph of f(x) translated three units right.

Important Observation

Unlike a vertical translation, a horizontal translation generally changes several coefficients when the equation is expanded.

Translating a Function to the Left

Suppose:

f(x) = 2x² − 3x + 4

Translate the graph two units to the left.

Step 1: Write the Transformation

g(x) = f(x + 2)

Step 2: Substitute

g(x) = 2(x + 2)² − 3(x + 2) + 4

Step 3: Expand

g(x) = 2(x² + 4x + 4) − 3x − 6 + 4

g(x) = 2x² + 8x + 8 − 3x − 6 + 4

Step 4: Simplify

g(x) = 2x² + 5x + 6

Interpretation

The graph has moved two units left.

Its shape and direction of opening remain unchanged.

Horizontal Translations in Vertex Form

Horizontal translations are particularly easy to describe when a quadratic is already in vertex form.

Consider:

f(x) = 3(x − 2)² + 4

The vertex is:

(2, 4)

Translate Right Five Units

New horizontal coordinate:

2 + 5 = 7

New function:

g(x) = 3(x − 7)² + 4

Translate Left Three Units

New horizontal coordinate:

2 − 3 = −1

New function:

h(x) = 3(x + 1)² + 4

Comparison

All three parabolas have the same shape and minimum value.

Only their horizontal positions differ.

Comparing Horizontal and Vertical Translations

Horizontal and vertical translations affect graphs in different ways.

Horizontal Translation

g(x) = f(x − h)

Changes the input.

Moves the graph left or right.

Vertical Translation

g(x) = f(x) + k

Changes the output.

Moves the graph up or down.

Comparison Table

Example

Parent function:

y = x²

Right three units:

y = (x − 3)²

Up three units:

y = x² + 3

These are different transformations.

Combining Horizontal and Vertical Translations

A quadratic function can be translated in both directions.

The general vertex form is:

y = a(x − h)² + k

Example

Consider:

y = (x − 3)² + 5

Compared with y = x²:

• Right three units.

• Up five units.

Vertex:

(3, 5)

Another Example

Consider:

y = (x + 4)² − 2

Compared with y = x²:

• Left four units.

• Down two units.

Vertex:

(−4, −2)

Important Observation

Horizontal and vertical translations can be combined without changing the shape of the parabola.

Comparing Translated Functions

Consider:

f(x) = x²

g(x) = (x − 3)²

h(x) = (x + 2)²

Comparison

Observations

All three graphs have identical shapes.

The graph of g(x) is three units right of f(x).

The graph of h(x) is two units left of f(x).

Therefore, g(x) is five units right of h(x).

Horizontal Translations and the Axis of Symmetry

The axis of symmetry passes vertically through the vertex.

For:

y = a(x − h)² + k

the axis of symmetry is:

x = h

Example

Consider:

y = (x − 5)² + 2

Axis of symmetry:

x = 5

Translate the graph four units left.

New axis of symmetry:

x = 1

New function:

y = (x − 1)² + 2

Important Observation

The axis of symmetry moves horizontally by the same amount as the parabola.

Horizontal Translations and the Range

Horizontal translations do not change the range of a quadratic function.

Example

Consider:

y = x² + 4

The minimum value is 4.

Range:

y ≥ 4

Translate the graph five units right:

y = (x − 5)² + 4

The minimum value remains 4.

Range:

y ≥ 4

Why?

Horizontal translations move points left or right but do not change their y-values.

The minimum or maximum value remains the same.

Horizontal Translations and Intercepts

Horizontal translations affect the intercepts of a quadratic graph.

Example

Consider:

y = x² − 4

The x-intercepts are:

x = −2 and x = 2

Translate the graph three units right.

New equation:

y = (x − 3)² − 4

Set y = 0:

(x − 3)² − 4 = 0

(x − 3)² = 4

x − 3 = ±2

Therefore:

x = 1 or x = 5

Interpretation

Both x-intercepts move three units right.

Important Observation

A horizontal translation moves every point on the graph, including its x-intercepts.

The y-intercept generally changes as well.

Identifying Translations from Equations

Sometimes two functions are given, and we must determine the translation between them.

Example

f(x) = (x − 2)² + 1

g(x) = (x − 6)² + 1

Original vertex:

(2, 1)

New vertex:

(6, 1)

Horizontal change:

6 − 2 = 4

Therefore:

g(x) is f(x) translated four units right.

Another Example

f(x) = −2(x + 3)² + 4

g(x) = −2(x − 1)² + 4

Original vertex:

(−3, 4)

New vertex:

(1, 4)

Horizontal change:

1 − (−3) = 4

Therefore:

g(x) is f(x) translated four units right.

Graphing Horizontal Translations Efficiently

When graphing a translated parabola, it is often unnecessary to calculate a completely new table of values.

Instead, translate known points.

Example

Parent function:

y = x²

Known points:

(−2, 4)

(−1, 1)

(0, 0)

(1, 1)

(2, 4)

Translate the graph three units right.

Add three to each x-coordinate.

Translated Points

The y-coordinates remain unchanged.

Graphing Steps

1. Identify the original vertex.

2. Determine the direction and distance of translation.

3. Move the vertex horizontally.

4. Translate several additional points.

5. Draw a smooth parabola through the points.

6. Label the new axis of symmetry.

Real-World Application: Projectile Motion

Quadratic functions can model the height of a projectile over time.

Example

Suppose the height of a ball is modelled by:

h(t) = −5(t − 2)² + 20

The vertex occurs at:

(2, 20)

This means the ball reaches a maximum height of 20 metres after two seconds.

Now consider:

g(t) = −5(t − 4)² + 20

The vertex is:

(4, 20)

Interpretation

The second height-time graph is translated two units right.

The maximum height remains 20 metres.

However, the time at which the maximum occurs increases from two seconds to four seconds.

Important Observation

A horizontal translation in a height-time graph represents a shift in time.

It does not represent physically moving the projectile sideways.

Real-World Application: Business Profit

Quadratic functions can represent profit relationships.

Example

Suppose a company's profit is modelled by:

P(x) = −2(x − 10)² + 200

The maximum profit is 200 units when x = 10.

Now consider:

Q(x) = −2(x − 15)² + 200

Interpretation

The second graph is translated five units right.

The maximum profit remains 200 units.

However, the quantity associated with maximum profit changes from 10 to 15.

Why Is This Useful?

Horizontal translations can represent changes in the input value at which an optimal outcome occurs.

Real-World Application: Engineering Design

Parabolic curves are used in bridge arches, reflectors, and architectural structures.

Example

An arch is modelled by:

y = −0.5x² + 8

The vertex is:

(0, 8)

Suppose an identical arch is constructed four metres to the right.

The new equation is:

y = −0.5(x − 4)² + 8

Interpretation

The highest point moves from:

(0, 8)

to:

(4, 8)

The arch retains the same height and shape.

Only its horizontal position changes.

Common Mistakes

Confusing the direction of translation

A negative sign inside the brackets moves the graph right.

A positive sign inside the brackets moves the graph left.

Changing the y-coordinate of the vertex

Horizontal translations change the x-coordinate, not the y-coordinate.

Changing the shape of the parabola

A horizontal translation does not stretch, compress, or reflect the graph.

Confusing horizontal and vertical translations

Horizontal translations affect the input.

Vertical translations affect the output.

Forgetting brackets in function notation

To translate f(x) right three units, use f(x − 3), not f(x) − 3.

Expanding incorrectly

Expressions such as (x − 3)² must be expanded carefully.

(x − 3)² = x² − 6x + 9

It is not x² − 9.

Confusing the vertex with the y-intercept

These are generally different points.

Assuming the range changes

Horizontal translations preserve the range.

Forgetting to move the axis of symmetry

The axis of symmetry moves horizontally with the vertex.

Changing y-values when translating points

For a horizontal translation, corresponding y-values remain unchanged.

Key Terms

• Horizontal Translation: Movement of a graph left or right without changing its shape.

• Transformation: Change in the position, size, or orientation of a graph.

• Parent Function: Simplest function in a family of related functions.

• Quadratic Function: Function whose highest variable power is two.

• Parabola: Graph of a quadratic function.

• Vertex: Maximum or minimum point of a parabola.

• Axis of Symmetry: Vertical line dividing a parabola into two matching halves.

• Vertex Form: Quadratic written as y = a(x − h)² + k.

• Horizontal Shift : Amount a graph moves left or right.

• Vertical Shift (k): Amount a graph moves up or down.

• Function Notation: Representation of a function using symbols such as f(x).

• Input: Value substituted into a function.

• Output: Value produced by a function.

• Domain: Set of possible input values.

• Range: Set of possible output values.

• Minimum Value: Lowest value of a function.

• Maximum Value: Highest value of a function.

• x-Intercept: Point where a graph crosses or touches the x-axis.

• y-Intercept: Point where a graph crosses the y-axis.

• Corresponding Points: Points occupying equivalent positions on original and transformed graphs.

• Translation Rule: Algebraic description of a graph's movement.

• Quadratic Model: Quadratic function used to represent a real-world relationship.

Key Takeaways

• Horizontal translations move parabolas left or right.

• The general rule is g(x) = f(x − h).

• Positive h-values produce rightward translations.

• Negative h-values produce leftward translations.

• Horizontal translations change the x-coordinate of the vertex.

• The y-coordinate of the vertex remains unchanged.

• The axis of symmetry moves horizontally with the vertex.

• The shape and direction of opening remain unchanged.

• The domain of a quadratic function remains all real numbers.

• The range remains unchanged.

• Horizontal translations can change the intercepts.

• Function notation describes horizontal translations by changing the input.

• A horizontal translation can be combined with a vertical translation.

• Comparing vertex coordinates helps identify the direction and distance of a translation.

• Horizontal translations are useful in physics, engineering, business, and mathematical modelling.

Check Your Understanding

Identifying Horizontal Translations

1. Describe the translation from y = x² to y = (x − 4)².

2. Describe the translation from y = x² to y = (x + 6)².

3. What happens to the vertex when a parabola moves five units right?

4. Does a horizontal translation change the shape of a parabola?

5. Does a horizontal translation change the range?

Finding Vertices

6. Find the vertex of y = (x − 3)².

7. Find the vertex of y = (x + 5)².

8. Find the vertex of y = (x − 2)² + 4.

9. Find the vertex of y = −2(x + 3)² − 1.

10. Describe the translation from y = (x − 2)² + 1 to y = (x − 7)² + 1.

Writing Equations

11. Translate y = x² four units right.

12. Translate y = x² three units left.

13. Translate y = (x − 2)² + 5 six units right.

14. Translate y = (x + 4)² − 3 two units left.

15. Write the general rule for a horizontal translation.

Comparing Graphs

16. Compare the vertices of y = (x − 3)² and y = (x + 2)².

17. Find the axis of symmetry of y = (x − 6)² + 4.

18. Find the range of y = (x + 5)² − 2.

19. Explain why horizontal translations change the x-intercepts.

20. Describe one real-world application of horizontal translations.

Answer Key

Extended Challenge

Identifying Transformations

1. Describe the translation from y = (x − 2)² to y = (x − 8)².

2. Describe the translation from y = (x + 5)² to y = (x + 1)².

3. Determine the translation from y = −3(x − 4)² + 2 to y = −3(x + 2)² + 2.

4. Explain why y = (x − 5)² and y = x² − 5 represent different transformations.

5. Describe how the vertex and axis of symmetry change when a parabola is translated seven units left.

Writing Equations

6. Translate y = x² − 4x + 3 two units right.

7. Translate y = 2x² + 3x − 1 three units left.

8. Translate y = −2(x − 3)² + 5 four units right.

9. Translate y = (x + 6)² − 2 five units left.

10. Write the equation obtained by translating y = x² eight units right and three units up.

Comparing Graphs

11. Compare y = (x − 4)² + 2 and y = (x + 3)² + 2.

12. Determine the vertices and axes of symmetry.

13. Explain why their ranges are identical.

14. Sketch both parabolas on the same coordinate grid.

15. Determine the horizontal distance between their vertices.

Applications

16. A projectile follows h(t) = −5(t − 2)² + 20. Write a new equation if the graph is translated three time units right.

17. A profit model is P(x) = −2(x − 10)² + 200. Write the equation after translating the graph five units left.

18. A parabolic arch is described by y = −0.5x² + 8. Write its equation after moving it six metres right.

19. Explain why horizontal translations preserve the maximum or minimum value.

20. Describe how horizontal translations can help compare mathematical models.

Final Investigation: Comparing Horizontal Translations

Consider the following quadratic functions:

Function A: y = x²

Function B: y = (x − 4)²

Function C: y = (x + 3)²

Function D: y = (x − 2)² + 5

Function E: y = (x + 5)² + 5

Complete the investigation.

1. Identify the parent quadratic function.

2. Describe the translation from A to B.

3. Describe the translation from A to C.

4. Determine the vertex of each function.

5. Determine the axis of symmetry of each function.

6. Identify which functions have the same range.

7. Create a table of values for Function A using x = −2, −1, 0, 1, 2.

8. Translate those points to create a table for Function B.

9. Explain how the x-coordinates of corresponding points are related.

10. Graph Functions A, B, and C on the same coordinate grid.

11. Compare their shapes.

12. Compare their ranges.

13. Determine the x-intercepts of Function B.

14. Determine the x-intercepts of Function C.

15. Describe the horizontal translation from Function D to Function E.

16. Calculate the horizontal distance between their vertices.

17. Graph Functions D and E on the same coordinate grid.

18. Explain why D and E have the same minimum value.

19. Write a general rule describing horizontal translations of quadratic functions.

20. Write a conclusion explaining how algebraic equations can be used to predict horizontal graph transformations.

3. Reflections

4. Stretching and Compressing

Stretching and Compressing Quadratic Functions

Introduction

Quadratic functions can be transformed in several ways, including translations, reflections, vertical stretches, and vertical compressions.

A vertical stretch makes the graph of a quadratic function appear narrower, while a vertical compression makes it appear wider.

These transformations are controlled by the coefficient a in the quadratic function.

Consider the parent function:

y = x²

Now compare:

• y = 3x² — narrower than the parent parabola.

• y = 0.5x² — wider than the parent parabola.

• y = −2x² — narrower and reflected across the x-axis.

Understanding vertical stretches and compressions helps us predict the shape of a parabola, compare quadratic functions, and graph transformed equations efficiently.

The Parent Quadratic Function

The parent quadratic function is:

f(x) = x²

Its graph is a U-shaped parabola.

Characteristics

• Vertex: (0, 0)

• Axis of symmetry: x = 0

• Direction of opening: Upward

• Minimum value: 0

• Domain: All real numbers

• Range: y ≥ 0

Table of Values

The parent function provides a reference for comparing other quadratic graphs.

Understanding the Coefficient a

The general quadratic function in vertex form is:

y = a(x − h)² + k

The coefficient a controls two important properties:

• The direction of opening.

• The vertical stretch or compression.

The magnitude |a| determines the vertical scale.

The sign of a determines whether the parabola opens upward or downward.

How a Affects the Graph

Important Observation

The coefficient a does not determine the location of the vertex when the equation is already in vertex form.

The values h and k determine the vertex.

Vertical Stretches

A vertical stretch occurs when the outputs of a function are multiplied by a factor greater than 1.

For the parent quadratic:

y = ax², where a > 1

The graph becomes narrower than y = x².

Example

Consider:

y = 3x²

Every y-value of the parent function is multiplied by 3.

Table of Values

Interpretation

At x = 1:

Parent function: y = 1

Stretched function: y = 3

At x = 2:

Parent function: y = 4

Stretched function: y = 12

The y-values are three times as far from the x-axis.

Important Observation

The vertex remains at (0, 0).

The axis of symmetry remains x = 0.

The parabola opens upward but appears narrower.

Why Does a Vertical Stretch Make a Parabola Narrower?

A vertical stretch increases the distance of each point from the x-axis.

Consider:

y = x²

and:

y = 4x²

For x = 1:

• Parent: y = 1

• Stretched: y = 4

For x = 2:

• Parent: y = 4

• Stretched: y = 16

The stretched parabola rises more rapidly as x moves away from the vertex.

This makes its sides appear steeper and the graph narrower.

Important Distinction

The transformation is called a vertical stretch because the y-values are multiplied.

It is not a horizontal compression, even though the graph appears narrower.

For a quadratic parabola, the two transformations can produce equivalent shapes under suitable scale factors, but they are described differently.

Vertical Compressions

A vertical compression occurs when the outputs of a function are multiplied by a factor between 0 and 1.

For the parent quadratic:

y = ax², where 0 < a < 1

The graph becomes wider than y = x².

Example

Consider:

y = 0.25x²

Every y-value of the parent function is multiplied by 0.25.

Table of Values

Interpretation

The y-values are one-quarter of the corresponding parent-function values.

The graph rises more slowly as x moves away from the vertex.

Important Observation

The vertex remains at (0, 0).

The axis of symmetry remains x = 0.

The parabola opens upward but appears wider.

Exploring Vertical Stretches and Compressions

Change the coefficient a to see how it affects the shape of the parabola.

Key Observation

Increasing |a| makes the parabola narrower.

Decreasing |a| toward zero makes the parabola wider.

Comparing Narrow and Wide Parabolas

Consider these functions:

f(x) = x²

g(x) = 4x²

h(x) = 0.25x²

Comparison

Observations

All three graphs share the same vertex.

All three graphs open upward.

The function y = 4x² rises more rapidly.

The function y = 0.25x² rises more slowly.

Important Conclusion

The magnitude of a determines the apparent width of a parabola.

Negative Coefficients and Reflections

The sign of a determines the direction of opening.

If a is negative, the parabola opens downward.

Example

Consider:

y = −2x²

The coefficient is:

a = −2

Since a is negative, the graph is reflected across the x-axis.

Since |a| = 2, the graph is vertically stretched by a factor of 2.

Comparison

Important Observation

The sign of a determines whether the graph opens upward or downward.

The magnitude |a| determines whether the graph is narrower or wider than the parent parabola.

Vertical Stretches in Vertex Form

Vertical stretches and compressions can be applied to translated parabolas.

Consider:

y = 3(x − 2)² + 4

The coefficient is:

a = 3

The vertex is:

(2, 4)

Interpretation

Compared with y = (x − 2)² + 4:

• The parabola is vertically stretched by a factor of 3.

• The vertex remains at (2, 4).

• The axis of symmetry remains x = 2.

• The parabola opens upward.

• The graph becomes narrower.

Important Observation

When the stretch is applied relative to the vertex form, the vertical distances from the vertex are multiplied by |a|.

The vertex itself remains unchanged.

Vertical Compressions in Vertex Form

Consider:

y = 0.5(x + 3)² − 2

The coefficient is:

a = 0.5

The vertex is:

(−3, −2)

Interpretation

Compared with y = (x + 3)² − 2:

• The parabola is vertically compressed by a factor of 0.5.

• The vertex remains at (−3, −2).

• The axis of symmetry remains x = −3.

• The parabola opens upward.

• The graph becomes wider.

Important Observation

The vertical compression changes the shape but does not change the vertex coordinates.

Graphing Stretched and Compressed Parabolas

A useful method is to begin with the vertex and calculate several nearby points.

Example

Graph:

y = 2(x − 1)² + 3

Step 1: Identify the Vertex

Vertex:

(1, 3)

Step 2: Identify the Coefficient

a = 2

The parabola opens upward and is narrower than the parent.

Step 3: Calculate Points

Step 4: Plot the Points

Plot:

(−1, 11)

(0, 5)

(1, 3)

(2, 5)

(3, 11)

Step 5: Draw the Parabola

Connect the points with a smooth curve.

The graph is symmetric about:

x = 1

Using the Step Pattern

A useful shortcut for graphing quadratic functions is the step pattern.

For the parent function y = x², moving one unit horizontally from the vertex changes the y-value by 1.

Moving another unit changes the y-value by an additional 3.

Moving another unit changes the y-value by an additional 5.

The pattern is:

1, 3, 5, 7, ...

For:

y = a(x − h)² + k

multiply these vertical changes by a.

Example

Consider:

y = 2(x − 1)² + 3

The step pattern becomes:

2, 6, 10, 14, ...

Starting from the vertex (1, 3):

• Move one unit right and two units up.

• Move another unit right and six units up.

• Move another unit right and ten units up.

Repeat symmetrically to the left.

Important Observation

If a is negative, the vertical steps move downward.

If |a| is less than 1, the vertical steps are smaller.

Comparing Stretches and Compressions Algebraically

Suppose:

f(x) = x²

and:

g(x) = 5x²

Then:

g(x) = 5f(x)

This represents a vertical stretch by a factor of 5.

Now suppose:

h(x) = 0.2x²

Then:

h(x) = 0.2f(x)

This represents a vertical compression by a factor of 0.2.

General Rule

If:

g(x) = af(x)

then:

• |a| > 1 produces a vertical stretch.

• 0 < |a| < 1 produces a vertical compression.

• a < 0 also produces a reflection across the x-axis.

Important Distinction

Multiplying an entire function by a changes all its output values.

Multiplying only the squared term in vertex form changes vertical distances from the vertex while preserving the vertex location.

Stretching and Compressing Translated Parabolas

Consider:

f(x) = (x − 2)² + 3

The vertex is:

(2, 3)

Vertical Stretch Relative to the Vertex

Change the coefficient of the squared term:

g(x) = 4(x − 2)² + 3

The vertex remains:

(2, 3)

Vertical Compression Relative to the Vertex

Use:

h(x) = 0.25(x − 2)² + 3

The vertex remains:

(2, 3)

Comparing the Functions

Important Observation

The coefficient a changes the shape of the parabola without changing its vertex when h and k remain constant.

Effect of a on the Range

The sign of a determines whether the vertex is a minimum or maximum.

Positive a

The parabola opens upward.

The vertex is a minimum.

For:

y = a(x − h)² + k, where a > 0

the range is:

y ≥ k

Negative a

The parabola opens downward.

The vertex is a maximum.

For:

y = a(x − h)² + k, where a < 0

the range is:

y ≤ k

Example

Consider:

y = −3(x − 2)² + 7

The vertex is:

(2, 7)

The parabola opens downward.

Maximum value:

7

Range:

y ≤ 7

Important Observation

Changing the magnitude of a does not change the range when the vertex and direction of opening remain unchanged.

Changing the sign of a reverses the direction of opening and changes the range.

Effect of a on Intercepts

Vertical stretches and compressions can affect the intercepts of a quadratic function.

Example

Consider:

y = x² − 4

The x-intercepts are:

x = −2 and x = 2

Now consider:

y = 4x² − 4

Set y = 0:

4x² − 4 = 0

4x² = 4

x² = 1

x = ±1

Interpretation

Increasing the coefficient of the squared term from 1 to 4 makes the parabola narrower.

The x-intercepts move closer to the axis of symmetry.

Important Observation

Changing a while keeping the vertex-form value k fixed can change the x-intercepts.

However, multiplying an entire function by a nonzero constant preserves its x-intercepts.

Real-World Application: Projectile Motion

Quadratic functions are used to model the height of projectiles.

Example

Consider two height-time models:

h₁(t) = −5(t − 2)² + 20

h₂(t) = −2(t − 2)² + 20

Both functions have the same vertex:

(2, 20)

Both reach a maximum height of 20 metres after two seconds.

However:

• h₁(t) has a larger |a| and is narrower.

• h₂(t) has a smaller |a| and is wider.

Interpretation

The first graph decreases more rapidly as time moves away from the vertex.

The second graph decreases more slowly.

Important Observation

In a physical projectile model, the coefficient of t² is related to acceleration.

Changing it may represent a different physical environment or mathematical scenario.

Real-World Application: Business Profit

Quadratic functions can represent profit relationships.

Example

Consider:

P₁(x) = −2(x − 10)² + 200

P₂(x) = −0.5(x − 10)² + 200

Both functions have the same maximum profit:

200

Both reach this maximum at:

x = 10

Comparison

P₁(x) is narrower.

P₂(x) is wider.

Interpretation

The first profit model decreases more rapidly as production moves away from the optimal quantity.

The second model decreases more gradually.

Important Observation

The magnitude of a can describe how sensitive an outcome is to changes in the input variable.

Real-World Application: Engineering Design

Parabolic curves are used in architecture and engineering.

Example

Consider two arch designs:

y = −0.5x² + 8

y = −0.2x² + 8

Both arches have a maximum height of eight metres.

However:

• The first arch is narrower.

• The second arch is wider.

Interpretation

Changing the coefficient a changes the curvature and width of the arch.

Engineers can use quadratic models to compare different structural designs.

Common Mistakes

Confusing stretches and compressions

A larger |a| makes a parabola narrower.

A smaller nonzero |a| makes it wider.

Assuming a negative coefficient always means compression

A negative sign indicates reflection.

The magnitude determines stretching or compression.

Ignoring the absolute value of a

Use |a| when comparing widths.

Changing the vertex unnecessarily

Changing only a in vertex form does not change the vertex.

Confusing vertical and horizontal transformations

Vertical stretches multiply output differences.

Horizontal transformations change input values.

Assuming all parabolas open upward

Negative a-values produce downward-opening parabolas.

Using incorrect step patterns

The basic step pattern must be multiplied by a.

Forgetting that a cannot equal zero in a quadratic function

If a = 0, the squared term disappears and the function is no longer quadratic.

Assuming the range always changes when a changes

If the vertex and direction of opening remain unchanged, the range remains unchanged.

Confusing narrowness with translation

A narrow parabola is not necessarily shifted horizontally or vertically.

Key Terms

• Vertical Stretch: Transformation that multiplies vertical distances by a factor greater than 1.

• Vertical Compression: Transformation that multiplies vertical distances by a factor between 0 and 1.

• Quadratic Function: Function whose highest variable power is two.

• Parabola: Graph of a quadratic function.

• Parent Function: Simplest function in a family of related functions.

• Leading Coefficient: Coefficient of the x² term.

• Coefficient a: Parameter controlling direction of opening and vertical scale.

• Absolute Value: Distance of a number from zero.

• Vertex: Maximum or minimum point of a parabola.

• Axis of Symmetry: Vertical line dividing a parabola into two matching halves.

• Reflection: Transformation producing a mirror image across a line.

• Transformation: Change in the position, size, or orientation of a graph.

• Vertex Form: Quadratic written as y = a(x − h)² + k.

• Standard Form: Quadratic written as y = ax² + bx + c.

• Narrow Parabola: Parabola with a larger |a| than a comparison parabola.

• Wide Parabola: Parabola with a smaller nonzero |a| than a comparison parabola.

• Step Pattern: Sequence of vertical changes used to graph a parabola.

• Minimum Value: Lowest value of a function.

• Maximum Value: Highest value of a function.

• Domain: Set of possible input values.

• Range: Set of possible output values.

• x-Intercept: Point where a graph crosses or touches the x-axis.

• y-Intercept: Point where a graph crosses the y-axis.

• Quadratic Model: Quadratic function used to represent a real-world relationship.

Key Takeaways

• The coefficient a controls the direction and vertical scale of a parabola.

• Positive a-values produce upward-opening parabolas.

• Negative a-values produce downward-opening parabolas.

• When |a| > 1, the parabola is vertically stretched relative to the parent function.

• When 0 < |a| < 1, the parabola is vertically compressed.

• Larger |a| values produce narrower parabolas.

• Smaller nonzero |a| values produce wider parabolas.

• The sign of a determines the direction of opening.

• The magnitude of a determines the vertical scale.

• Changing only a in vertex form does not change the vertex.

• The axis of symmetry remains unchanged when only a changes.

• The step pattern can be multiplied by a to graph transformed parabolas.

• Vertical stretches and compressions can be combined with reflections and translations.

• The range depends on the vertex and direction of opening.

• Quadratic transformations are useful in physics, engineering, business, and mathematical modelling.

Check Your Understanding

Identifying Stretches and Compressions

1. Is y = 4x² narrower or wider than y = x²?

2. Is y = 0.2x² narrower or wider than y = x²?

3. What does a = −3 indicate?

4. What does a = −0.5 indicate?

5. Which is narrower: y = 2x² or y = 5x²?

Comparing Graphs

6. Compare the directions of opening of y = 3x² and y = −3x².

7. Which is wider: y = 0.25x² or y = 0.75x²?

8. Find the vertex of y = 4(x − 2)² + 3.

9. Find the axis of symmetry of y = −2(x + 5)² − 1.

10. Explain why changing a in vertex form does not change the vertex.

Graphing

11. Calculate y when x = 2 for y = 3x².

12. Calculate y when x = 4 for y = 0.25x².

13. Find the first three vertical step changes for y = 2x².

14. Find the first three vertical step changes for y = −3x².

15. Describe how to graph y = 0.5(x − 1)² + 2.

Applications

16. Explain why a larger |a| produces a narrower parabola.

17. Find the range of y = 3(x − 2)² + 5.

18. Find the range of y = −0.5(x + 1)² + 4.

19. Explain how changing a affects the shape of a profit graph.

20. Describe one real-world application of vertical stretches or compressions.

Answer Key

Extended Challenge

Identifying Transformations

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

2. Describe the transformation from y = x² to y = 0.125x².

3. Describe the transformation from y = x² to y = −4x².

4. Describe the transformation from y = x² to y = −0.25x².

5. Explain why y = −3x² and y = 3x² have the same width.

Comparing Parabolas

6. Compare y = 2x² and y = 0.5x².

7. Determine which graph is narrower.

8. Calculate the y-values at x = 2 for both functions.

9. Explain how the coefficient affects the rate of increase.

10. Sketch both graphs on the same coordinate grid.

Vertex Form

11. Find the vertex of y = 5(x − 3)² + 2.

12. Find the vertex of y = −0.5(x + 4)² − 1.

13. Compare the widths of y = 3(x − 2)² + 4 and y = 0.25(x − 2)² + 4.

14. Explain why both functions in Question 13 have the same vertex.

15. Determine the ranges of both functions.

Applications

16. Compare h₁(t) = −5(t − 2)² + 20 and h₂(t) = −2(t − 2)² + 20.

17. Explain which height-time graph is narrower.

18. Compare P₁(x) = −3(x − 10)² + 200 and P₂(x) = −0.5(x − 10)² + 200.

19. Explain which profit model is more sensitive to changes in production.

20. Describe how engineers might use different a-values when designing parabolic arches.

Final Investigation: Comparing Vertical Stretches and Compressions

Consider the following quadratic functions:

Function A: y = x²

Function B: y = 3x²

Function C: y = 0.25x²

Function D: y = −2x²

Function E: y = 0.5(x − 2)² + 3

Complete the investigation.

1. Identify the parent quadratic function.

2. Determine the coefficient a for each function.

3. Identify which functions open upward.

4. Identify which functions open downward.

5. Identify which functions are vertically stretched relative to the parent.

6. Identify which functions are vertically compressed relative to the parent.

7. Determine the vertex of each function.

8. Determine the axis of symmetry of each function.

9. Create a table of values for Function A using x = −2, −1, 0, 1, 2.

10. Create corresponding tables for Functions B, C, and D.

11. Graph Functions A, B, C, and D on the same coordinate grid.

12. Compare the widths of the four graphs.

13. Explain why Function B is narrower than Function A.

14. Explain why Function C is wider than Function A.

15. Explain how Function D combines a reflection and stretch.

16. Determine the vertex and axis of symmetry of Function E.

17. Graph Function E.

18. Explain how Function E differs from the parent function.

19. Write a general rule describing the effect of coefficient a.

20. Write a conclusion explaining how vertical stretches, compressions, reflections, and translations can be combined to transform quadratic functions.

5. Combining Transformations

Combining Transformations of Quadratic Functions

Introduction

Quadratic functions can undergo several transformations at the same time. These transformations include horizontal translations, vertical translations, reflections, vertical stretches, and vertical compressions.

Rather than studying each transformation separately, we can combine them to create many different parabolas from the parent quadratic function:

y = x²

For example, consider:

y = −2(x − 3)² + 4

This function includes several transformations:

• A reflection across the x-axis.

• A vertical stretch by a factor of 2.

• A horizontal translation 3 units right.

• A vertical translation 4 units up.

By recognising these changes, we can predict the graph's shape, position, direction of opening, vertex, and axis of symmetry before plotting any points.

The ability to combine transformations is important for graphing functions efficiently, interpreting mathematical models, and understanding relationships between different quadratic equations.

Reviewing the Parent Quadratic Function

The parent quadratic function is:

f(x) = x²

Its graph is a U-shaped parabola.

Characteristics

• Vertex: (0, 0)

• Axis of symmetry: x = 0

• Direction of opening: Upward

• Minimum value: 0

• Domain: All real numbers

• Range: y ≥ 0

The parent function provides a starting point for analysing transformed quadratic functions.

Table of Values

Every transformed parabola can be related to this parent function through changes in its equation.

Understanding Vertex Form

The most useful form for identifying multiple transformations is:

y = a(x − h)² + k

Each parameter has a specific effect.

The vertex is:

(h, k)

The axis of symmetry is:

x = h

Example

Consider:

y = −3(x + 2)² + 5

Identify the parameters:

a = −3

h = −2

k = 5

Transformations

• Reflect across the x-axis.

• Vertically stretch by a factor of 3.

• Translate 2 units left.

• Translate 5 units up.

Graph Characteristics

• Vertex: (−2, 5)

• Axis of symmetry: x = −2

• Opens downward

• Maximum value: 5

• Range: y ≤ 5

Important Observation

The vertex and direction of opening can be determined without expanding the equation.

Identifying Multiple Transformations

Consider:

y = 2(x − 4)² − 3

Horizontal Translation

The expression (x − 4) indicates a shift 4 units right.

Vertical Translation

The −3 indicates a shift 3 units down.

Vertical Stretch

The coefficient 2 indicates a vertical stretch by a factor of 2.

Reflection

Since a is positive, there is no reflection across the x-axis.

Final Characteristics

• Vertex: (4, −3)

• Axis of symmetry: x = 4

• Opens upward

• Narrower than the parent parabola

• Minimum value: −3

Important Observation

Each part of vertex form provides information about a different transformation.

Describing Transformations in the Correct Order

When combining transformations, the order matters because some transformations do not commute.

A reliable sequence for constructing:

y = a(x − h)² + k

from the parent function y = x² is:

1. Apply the horizontal translation h.

2. Apply the vertical stretch or compression by |a|.

3. Reflect across the x-axis if a is negative.

4. Apply the vertical translation k.

This sequence is particularly useful because it produces the equation directly from vertex form.

Example

Construct:

y = −2(x − 3)² + 4

Start with:

y = x²

Step 1: Translate Right 3

y = (x − 3)²

Step 2: Vertically Stretch by 2

y = 2(x − 3)²

Step 3: Reflect Across the x-Axis

y = −2(x − 3)²

Step 4: Translate Up 4

y = −2(x − 3)² + 4

Important Observation

The vertical translation is applied after the vertical stretch and reflection.

Otherwise, the vertical position may also be multiplied or reflected.

Why the Order of Transformations Matters

Consider the parent function:

f(x) = x²

Suppose we want to reflect the graph across the x-axis and translate it upward by 3 units.

Method A: Reflect, Then Translate

Reflect:

y = −x²

Translate upward by 3:

y = −x² + 3

Vertex:

(0, 3)

Method B: Translate, Then Reflect

Translate upward by 3:

y = x² + 3

Reflect the entire function:

y = −(x² + 3)

Simplify:

y = −x² − 3

Vertex:

(0, −3)

Comparison

The final graphs are different.

Important Conclusion

The order of vertical translations and reflections can change the final graph.

However, some transformations can be performed in either order without changing the result. For example, horizontal and vertical translations commute.

Predicting the Graph Before Plotting

Before graphing a transformed quadratic, examine its equation.

Consider:

y = −0.5(x + 4)² + 6

Step 1: Identify the Vertex

h = −4

k = 6

Vertex:

(−4, 6)

Step 2: Determine the Direction of Opening

a = −0.5

Since a is negative, the parabola opens downward.

Step 3: Determine the Width

|a| = 0.5

Since 0 < |a| < 1, the parabola is wider than the parent.

Step 4: Determine the Axis of Symmetry

x = −4

Step 5: Determine the Range

The vertex is a maximum.

y ≤ 6

Prediction

The graph will be a wide, downward-opening parabola with its highest point at (−4, 6).

This prediction can be made before calculating any additional points.

Exploring Combined Transformations

Adjust the parameters to see how the graph changes.