4. Transformations of Polynomial Functions

Transformations of Polynomial Functions

A transformation changes the position, orientation, or shape of a graph without changing the basic family of function it belongs to.

For example, begin with the basic quadratic function:

f(x) = x²

We can move it up, down, left, or right. We can reflect it or stretch and compress it.

Examples include:

y = x² + 3

y = (x − 2)²

y = −x²

y = 3x²

Each equation is a transformation of the original function.

Learning Targets

• I can identify vertical and horizontal translations.
• I can identify reflections and stretches.
• I can determine the effect of transformations on polynomial graphs.
• I can write transformed polynomial equations.
• I can sketch transformed polynomial functions.

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Parent Functions

A parent function is the simplest version of a particular family of functions.

Some common polynomial parent functions are:

Linear: f(x) = x

Quadratic: f(x) = x²

Cubic: f(x) = x³

Quartic: f(x) = x⁴

Transformations create new functions based on these basic graphs.

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The General Transformation Form

A transformed function can often be written as:

g(x) = a f(b(x − h)) + k

Each value controls a different transformation.

h controls horizontal translation.

k controls vertical translation.

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

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

Understanding these values allows us to predict a graph without plotting many individual points.

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Vertical Translations

A vertical translation moves a graph upward or downward.

Starting with:

y = f(x)

the transformed function:

y = f(x) + k

moves the graph vertically.

If:

k > 0

the graph moves up.

If:

k < 0

the graph moves down.

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Moving a Graph Up

Consider:

f(x) = x²

Now transform it to:

g(x) = x² + 3

Every y-value increases by 3.

Therefore, the entire graph moves:

3 units upward

The original vertex:

(0, 0)

becomes:

(0, 3)

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Moving a Graph Down

Consider:

f(x) = x²

and:

g(x) = x² − 4

Every y-value decreases by 4.

Therefore, the graph moves:

4 units downward

The vertex moves from:

(0, 0)

to:

(0, −4)

The shape and orientation of the graph remain unchanged.

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Horizontal Translations

Horizontal translations move graphs left or right.

Starting with:

y = f(x)

the transformation:

y = f(x − h)

moves the graph horizontally.

The direction can initially seem backwards.

f(x − h) → right by h

f(x + h) → left by h

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Moving a Graph Right

Consider:

f(x) = x²

and:

g(x) = (x − 3)²

The graph moves:

3 units right

The vertex changes from:

(0, 0)

to:

(3, 0)

Notice that the equation contains:

x − 3

but the graph moves right.

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Moving a Graph Left

Consider:

g(x) = (x + 2)²

We can think of this as:

g(x) = (x − (−2))²

Therefore, the graph moves:

2 units left

The vertex becomes:

(−2, 0)

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Combining Horizontal and Vertical Translations

Transformations can occur together.

Consider:

g(x) = (x − 2)² + 3

Starting with:

f(x) = x²

the graph moves:

2 units right

and:

3 units up

The vertex moves from:

(0, 0)

to:

(2, 3)

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Worked Example — Identifying a Translation

Describe the transformation:

g(x) = (x + 4)² − 2

Start with:

f(x) = x²

Inside the brackets:

x + 4

means:

4 units left

Outside the brackets:

−2

means:

2 units down

Therefore:

Left 4 and down 2

The vertex moves to:

(−4, −2)

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Reflections Across the X-Axis

Multiplying the entire function by −1 reflects its graph across the x-axis.

Starting with:

y = f(x)

the reflection is:

y = −f(x)

For example:

f(x) = x²

becomes:

g(x) = −x²

Every point:

(x, y)

becomes:

(x, −y)

The upward-opening parabola becomes a downward-opening parabola.

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Reflection of a Cubic Function

Consider:

f(x) = x³

Reflecting across the x-axis gives:

g(x) = −x³

The original graph has:

Left end → down

Right end → up

After reflection:

Left end → up

Right end → down

The entire graph has been flipped vertically.

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Reflections Across the Y-Axis

Replacing x with −x reflects a graph across the y-axis.

Starting with:

y = f(x)

the transformed function is:

y = f(−x)

For example, consider:

f(x) = (x − 2)³

Then:

f(−x) = (−x − 2)³

produces a reflection across the y-axis.

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Some Reflections Appear Unchanged

Consider:

f(x) = x²

Reflect across the y-axis:

f(−x) = (−x)²

But:

(−x)² = x²

Therefore, the graph looks unchanged.

This happens because:

f(x) = x²

is symmetric about the y-axis.

The same is true for:

x⁴, x⁶, x⁸, ...

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Vertical Stretches

Consider:

y = a f(x)

If:

|a| > 1

the graph undergoes a vertical stretch.

For example:

f(x) = x²

and:

g(x) = 3x²

Every y-value is multiplied by 3.

A point:

(1, 1)

becomes:

(1, 3)

A point:

(2, 4)

becomes:

(2, 12)

For a parabola, this makes the graph appear narrower or steeper.

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Vertical Compressions

If:

0 < |a| < 1

the graph undergoes a vertical compression.

For example:

g(x) = 0.5x²

Every y-value is multiplied by:

0.5

The graph appears wider or flatter than:

y = x²

The basic shape remains recognizable.

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The Sign and Size of a

In:

g(x) = a f(x)

the value of a tells us two things.

If:

a < 0

the graph is reflected across the x-axis.

If:

|a| > 1

there is a vertical stretch.

If:

0 < |a| < 1

there is a vertical compression.

For example:

g(x) = −3x²

contains both:

Reflection across the x-axis

and:

Vertical stretch by factor 3

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Worked Example — Reflection and Stretch

Describe:

g(x) = −2(x − 3)² + 5

Compare it with:

f(x) = x²

The:

−2

means:

Reflection across the x-axis

and:

Vertical stretch by factor 2

The:

x − 3

means:

3 units right

The:

+5

means:

5 units up

Therefore, the transformed parabola has its vertex at:

(3, 5)

and opens downward.

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Horizontal Stretches and Compressions

Changes inside the function affect horizontal dimensions.

Consider:

g(x) = f(bx)

If:

|b| > 1

the graph undergoes a horizontal compression by factor:

1/|b|

If:

0 < |b| < 1

the graph undergoes a horizontal stretch.

The horizontal effect is reciprocal, which can make these transformations more difficult to recognize.

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Worked Example — Horizontal Compression

Suppose:

f(x) = x³ − x

and:

g(x) = f(2x)

The input has been multiplied by:

2

Therefore, the graph is horizontally compressed by a factor of:

1/2

Features of the graph occur at half their original x-coordinates.

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Worked Example — Horizontal Stretch

Suppose:

g(x) = f(x/3)

This can be written:

g(x) = f((1/3)x)

Since:

b = 1/3

the graph is horizontally stretched by a factor of:

3

Features become three times farther from the y-axis.

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Combining Several Transformations

Consider:

g(x) = −2(x + 1)³ + 4

Start with:

f(x) = x³

Now identify each transformation.

x + 1

means:

1 unit left

The factor:

−2

means:

Reflection across the x-axis

and:

Vertical stretch by factor 2

Finally:

+4

means:

4 units up

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Transforming Important Points

Instead of calculating many new points, we can transform points from the original graph.

For a transformation:

g(x) = a f(x − h) + k

a point:

(x, y)

on the original graph moves to:

(x + h, ay + k)

This provides a powerful way to sketch transformed functions.

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Worked Example — Transforming Points

Suppose:

f(x) = x²

and:

g(x) = 2(x − 3)² + 1

A point on the parent function is:

(0, 0)

Apply the transformation.

Move right 3:

x = 3

Multiply the y-value by 2:

2(0) = 0

Move up 1:

y = 1

So:

(0, 0) → (3, 1)

Another parent point is:

(1, 1)

It becomes:

(4, 3)

because:

x: 1 + 3 = 4

and:

y: 2(1) + 1 = 3

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Transformations and Zeros

Transformations can change the zeros of a polynomial.

Consider:

f(x) = x² − 4

Its zeros are:

x = −2 and x = 2

Now move the graph upward:

g(x) = x² − 4 + 4

which simplifies to:

g(x) = x²

The two original x-intercepts have moved until they meet at:

x = 0

The transformed graph now has one distinct real zero with multiplicity 2.

Transformations can therefore change where a graph intersects the axes.

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Horizontal Translation of Zeros

Consider:

f(x) = (x − 1)(x + 2)

Its zeros are:

x = 1 and x = −2

Now translate the graph 3 units right:

g(x) = f(x − 3)

Every important feature moves 3 units right.

Therefore, the zeros become:

x = 4 and x = 1

This allows us to predict transformed roots without solving the new polynomial from the beginning.

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Transformations and End Behaviour

Some transformations change end behaviour while others do not.

Translations do not change the fundamental end directions.

For example, moving a positive quartic left, right, up, or down still leaves:

Both ends → up

A reflection across the x-axis reverses the vertical end behaviour.

A positive quartic with:

Both ends → up

becomes a negative quartic with:

Both ends → down

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Transformations and Multiplicity

Translations and stretches do not normally change the multiplicity pattern of roots.

If a graph originally touches and turns at a double root, a horizontal translation moves that behaviour to a new x-coordinate.

If a graph crosses and flattens at a root with multiplicity 3, translating the graph moves that feature without changing its basic crossing behaviour.

This makes transformations useful when sketching complicated polynomial functions from simpler ones.

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Writing a Transformed Equation

Suppose:

f(x) = x²

and you are asked to:

• Move right 4
• Move down 2

Start with:

x²

Right 4 means replace x with:

x − 4

giving:

(x − 4)²

Down 2 means subtract 2:

(x − 4)² − 2

Therefore:

g(x) = (x − 4)² − 2

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Worked Example — Writing from a Description

Start with:

f(x) = x³

Apply:

• Reflection across the x-axis
• Vertical stretch by factor 3
• Move left 2
• Move up 5

Left 2:

(x + 2)³

Vertical stretch:

3(x + 2)³

Reflect:

−3(x + 2)³

Move up:

−3(x + 2)³ + 5

Therefore:

g(x) = −3(x + 2)³ + 5

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Sketching Transformed Polynomial Functions

A good sketching process is:

• Identify the parent function.
• Identify horizontal translations.
• Identify vertical translations.
• Identify reflections.
• Identify stretches or compressions.
• Locate important transformed points.
• Determine end behaviour.
• Draw a smooth curve consistent with all of these features.

You usually do not need to create a large table of values.

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Worked Example — Sketching a Transformed Quadratic

Sketch:

g(x) = −(x − 2)² + 4

Start with:

f(x) = x²

Transformations:

Right 2

Reflect across the x-axis

Up 4

The original vertex:

(0, 0)

moves to:

(2, 4)

Because of the reflection, the parabola opens downward.

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Worked Example — Sketching a Transformed Cubic

Consider:

g(x) = 2(x + 1)³ − 3

Start with:

f(x) = x³

Transformations:

Left 1

Vertical stretch by factor 2

Down 3

The central point:

(0, 0)

moves to:

(−1, −3)

The graph still has the general increasing cubic shape because there is no reflection.

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Comparing Original and Transformed Graphs

A transformation is often easier to understand when the original and transformed graphs are shown together.

For example, compare:

f(x) = x²

with:

g(x) = −2(x − 3)² + 4

The second graph is:

• Shifted right
• Shifted up
• Reflected
• Vertically stretched

Yet it remains recognizable as a transformed quadratic.

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Reading Transformations from an Equation

Consider:

g(x) = 4(x − 5)⁴ − 2

Identify each part.

4

means:

Vertical stretch by factor 4

x − 5

means:

5 units right

−2

means:

2 units down

The parent function is:

f(x) = x⁴

Because the coefficient 4 is positive, there is no reflection across the x-axis.

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Reading Transformations from a Graph

You may also be given a parent graph and a transformed graph.

Look for:

• Has the graph moved horizontally?
• Has it moved vertically?
• Has it flipped?
• Has it become steeper or flatter?
• Have important points moved?
• Has the end behaviour reversed?

Comparing corresponding features can reveal the transformations.

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A Useful Transformation Summary

f(x) + k

moves the graph:

up k units

f(x) − k

moves the graph:

down k units

f(x − h)

moves the graph:

right h units

f(x + h)

moves the graph:

left h units

−f(x)

reflects the graph:

across the x-axis

f(−x)

reflects the graph:

across the y-axis

a f(x), |a| > 1

produces a:

vertical stretch

a f(x), 0 < |a| < 1

produces a:

vertical compression

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Why Horizontal Changes Seem Backwards

One of the most common difficulties with transformations is horizontal translation.

Why does:

f(x − 3)

move the graph right rather than left?

Suppose an important feature originally occurs when:

x = 0

For the transformed function, the same input to f occurs when:

x − 3 = 0

Therefore:

x = 3

The feature has moved from:

x = 0

to:

x = 3

So the graph moves right.

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Polynomial Transformations in Real Applications

Transformations are useful whenever a mathematical model needs to be adjusted.

They can represent:

• Changes in starting position
• Changes in baseline values
• Reversed relationships
• Increased or decreased scale
• Shifts in experimental data
• Adjustments to mathematical models
• Changes in computer-generated curves

The same transformation ideas are also fundamental in physics, engineering, computer graphics, and data analysis.

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Common Mistakes

Mistake: Reversing Horizontal Translation

f(x − 3) means:

Right 3

not left 3.

f(x + 3) means:

Left 3

not right 3.

Mistake: Confusing Horizontal and Vertical Changes

Changes outside the function generally affect vertical position or scale.

Changes inside the function generally affect horizontal position or scale.

Mistake: Missing a Reflection

In:

−(x − 2)²

the negative sign outside the brackets reflects the graph across the x-axis.

Mistake: Treating a Vertical Stretch as a Translation

In:

3x²

the graph is not moved upward by 3.

Its y-values are multiplied by 3.

Mistake: Applying Horizontal Scale Directly

For:

f(2x)

the horizontal scale factor is:

1/2

not 2.

Mistake: Forgetting That Transformations Can Combine

A function such as:

−2(x + 3)⁴ + 5

contains several transformations at once.

Analyze each part separately.

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Key Terms

Transformation – a change to the position, orientation, or scale of a graph.

Parent function – the simplest function in a particular family.

Translation – movement of a graph without changing its shape.

Vertical translation – movement upward or downward.

Horizontal translation – movement left or right.

Reflection – flipping a graph across an axis.

Vertical stretch – multiplying y-values so they move farther from the x-axis.

Vertical compression – multiplying y-values so they move closer to the x-axis.

Horizontal stretch – spreading a graph horizontally.

Horizontal compression – squeezing a graph horizontally.

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Check Your Understanding

1. Describe the transformation from:

f(x) = x²

to:

g(x) = x² + 5

2. Describe:

g(x) = (x − 4)²

3. Describe:

g(x) = (x + 3)² − 2

4. What transformation occurs in:

g(x) = −x³

5. Describe the transformations in:

g(x) = 3x²

6. Describe the transformations in:

g(x) = −2x⁴

7. Describe:

g(x) = 2(x − 3)² + 5

8. Describe:

g(x) = −3(x + 2)³ − 4

9. Write an equation obtained by moving:

f(x) = x²

4 units right and 6 units up.

10. Write an equation obtained by moving:

f(x) = x³

2 units left and 5 units down.

11. Write an equation for a reflection of:

f(x) = x⁴

across the x-axis.

12. Write an equation for a vertical stretch of:

f(x) = x²

by a factor of 5.

13. Explain why:

f(x − 3)

moves a graph right rather than left.

14. Describe the effect of:

g(x) = f(2x)

on the graph of f(x).

15. Explain how you would sketch:

g(x) = −2(x − 1)⁴ + 3

starting from the graph of:

f(x) = x⁴

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Key Takeaways

• Transformations change the position, orientation, or scale of polynomial graphs.
• f(x) + k moves a graph vertically.
• f(x − h) moves a graph horizontally.
• Horizontal translations appear to work in the opposite direction from the sign inside the brackets.
• −f(x) reflects a graph across the x-axis.
• f(−x) reflects a graph across the y-axis.
• Multiplying a function by a number greater than 1 produces a vertical stretch.
• Multiplying by a number between 0 and 1 produces a vertical compression.
• Changes inside the function can create horizontal stretches or compressions.
• Several transformations can occur in the same equation.
• Important points can be transformed instead of recalculated from the beginning.
• Transformations can affect zeros and intercepts.
• Reflections can change end behaviour.
• Translations do not change the fundamental degree or basic polynomial family.
• Recognizing transformations allows complicated polynomial graphs to be sketched efficiently.

Did You Know?

Graph transformations are not limited to algebra. The same basic ideas—translation, reflection, stretching, and compression—are used extensively in computer graphics and animation. When an object on a screen is moved, flipped, or resized, the software performs mathematical transformations closely related to the transformations used to move and reshape function graphs.