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.