Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (2024)

This lesson will discuss how to apply different transformations to quadratic functions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Transforming the Graph of a Function

The graph of the quadratic function f(x)=-x2+2x+1 is drawn on the coordinate plane. By applying transformations to the parabola that corresponds to f, draw the graph of g(x)=-x2+4x.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (1)

Discussion

Reflection of Quadratic Functions

A reflection of a function is a transformation that flips a graph over a line called the line of reflection. A reflection in the x-axis is achieved by changing the sign of every output value of the function rule. In other words, the sign of the y-coordinate of every point on the graph of a function should be changed. Consider the quadratic parent function y=x2.

Functiony=x2Reflectioninthex-axisy=-x2

The reflection of the corresponding parabola can be shown on a coordinate plane.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (2)

A reflection in the y-axis is instead achieved by changing the sign of every input value. However, since (-x)2 is equivalent to x2, reflecting y=x2 in the y-axis does not change the graph. For this reason, the reflection of the graph of another quadratic function will be shown.

Functiony=(x1)2Reflectioninthey-axisy=(-x1)2

This transformation can also be shown on a coordinate plane.

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Example

Reflecting a Quadratic Function

Kriz is given an extra credit math assignment about quadratic functions and parabolas. The assignment consists of four tasks. For the first task, Kriz is given the graph of the function y=21(x+2)2+1.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (4)

Help Kriz with their extra credit assignment.

a Reflect the given parabola in the x-axis and write its corresponding equation.

b Reflect the given parabola in the y-axis and write its corresponding equation.

Answer

a Graph:

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (5)

Equation: y=-21(x+2)21

b Graph:

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (6)

Equation: y=21(x2)2+1

Hint

a Recall that the graph of y=-f(x) is a reflection of the graph of y=f(x) in the x-axis.

b The graph of y=f(-x) is a reflection of the graph of y=f(x) in the y-axis.

Solution

a The reflection of the graph of y=f(x) in the x-axis is given by the equation of y=-f(x).

GivenFunctionf(x)=21(x+2)2+1Reflectioninthex-axis-f(x)=-(21(x+2)2+1)-f(x)=-21(x+2)21

Therefore, the graph of y=-21(x+2)21 is a reflection of the graph of y=21(x+2)2+1 in the x-axis.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (7)

b The reflection of the graph of y=f(x) in the y-axis is given by the equation of y=f(-x).

GivenFunctionf(x)=21(x+2)2+1Reflectioninthey-axisf(-x)=21(-x+2)2+1

To obtained equation can be simplified.

f(-x)=21(-x+2)2+1

f(-x)=21(-(-x+2))2+1

Distr

Distribute -1

f(-x)=21(x2)2+1

Therefore, the graph of y=21(x2)2+1 is a reflection of the graph of y=21(x+2)2+1 in the y-axis.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (8)

Explore

Stretching and Shrinking a Parabola

In the coordinate plane, the parabola that corresponds to the quadratic function y=af(bx) can be seen. Observe how the graph is vertically and horizontally stretched and shrunk by changing the values of a and b.

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Discussion

Stretch and Shrink of Quadratic Functions

A function graph is vertically stretched or shrunk by multiplying the output of a function rule by some constant a, where a>0. This constant must be positive, otherwise a reflection is involved. Consider the quadratic function y=x2+1.

Functiony=x2+1VerticalStretch/ShrinkbyaFactorofay=a(x2+1)

If a is greater than 1, the parabola is vertically stretched by a factor of a. Conversely, if a is less than 1, the graph is vertically shrunk by a factor of a. If a=1, then there is no stretch nor shrink. Here, the y-coordinates of all points on the graph are multiplied by the factor a.

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Similarly, a function graph is horizontally stretched or shrunk by multiplying the input of a function rule by some constant b, where b>0. Again, the constant must be positive because if it was negative, a reflection would be required.

Functiony=x2+1HorizontalStretch/ShrinkbyaFactorofby=(bx)2+1

In this case, if b is greater than 1, the graph is horizontally shrunk by a factor of b. Conversely, if b is less than 1, the graph is horizontally stretched by a factor of b. If b=1, then there is neither a stretch nor shrink of the graph. Here, the x-coordinates of all points on the graph are multiplied by the factor b1.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (11)

The information about vertical and horizontal stretches and shrinks of the graph of a function f can be summed up in a table.

Vertical Horizontal
Stretch af(x), with a>1 f(ax), with 0<a<1
Shrink af(x), with 0<a<1 f(ax), with a>1

Kriz's second task is about horizontal and vertical stretches and shrinks of parabolas.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (12)

They are given the quadratic function y=x23 and want to write the function rules of two related functions.

a A quadratic function whose graph is a vertical stretch of the graph of y=x23 by a factor of 2.

b A quadratic function whose graph is a horizontal shrink of the graph of y=x23 by a factor of 4.

Hint

a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant.

b A function graph is horizontally stretched or shrunk by multiplying the input of a function rule by a positive constant.

Solution

a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant. If the constant is greater than 1, the graph is vertically stretched. Therefore, if the graph of the given function is to be vertically stretched by a factor of 2, the function rule must be multiplied by 2.

Functiony=x23VerticalStretchbyaFactorof2y=2(x23)

Recall that stretching a graph by a factor of 2 means multiplying the y-coordinates of all the points on the curve by a factor of 2. This can be seen in a coordinate plane.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (13)

It is worth noting that the resulting function can be simplified by distributing 2.

y=2(x23)y=2x26

b A function graph is horizontally stretched or shrunk by multiplying the input of the function rule by a positive constant. If the constant is greater than 1, the graph is horizontally shrunk. Therefore, to horizontally shrink the graph by a factor of 4, the input of the function rule must be multiplied by 4.

Functiony=x23HorizontalShrinkbyaFactorof4y=(4x)23

This can be seen in a coordinate plane.

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It is worth noting that the resulting function can be simplified by using the Power of a Product Property.

y=(4x)23y=16x23

Pop Quiz

Stating the Factor of a Stretch or a Shrink

The graph of the quadratic function y=x2 is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.

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Example

Combining Transformations of Quadratic Functions

Kriz's assignment is getting more interesting, as the third task is about combining reflections with vertical and horizontal stretches and shrinks.

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This time, they are given the quadratic function y=(x1)2 and want to write the function rules of two other functions.

a A quadratic function whose graph is a vertical stretch by a factor of 3 followed by a reflection in the y-axis of the graph of y=(x1)2.

b A quadratic function whose graph is a horizontal shrink by a factor of 2 followed by a reflection in the x-axis of the graph of y=(x1)2.

Hint

a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant. Furthermore, a function graph is reflected in the y-axis by changing the sign of the input.

b A function graph is horizontally stretched or shrunk by multiplying the input of the function rule by a positive constant. Furthermore, a function graph is reflected in the x-axis by changing the sign of the output.

Solution

a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant. If the constant is greater than 1, the graph is vertically stretched. Therefore, if the graph of the given function is to be vertically stretched by a factor of 3, the function rule must be multiplied by 3.

Functiony=(x1)2VerticalStretchbyaFactorof3y=3(x1)2

Furthermore, a function graph is reflected in the y-axis by changing the sign of the input.

Functiony=3(x1)2Reflectioninthey-axisy=3(-x1)2

These transformations are illustrated by the following diagram.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (17)

The resulting function can be simplified by squaring the binomial and then distributing the 3.

y=3(-x1)2

Simplify right-hand side

y=3(-(-x1))2

Distr

Distribute -1

y=3(x+1)2

ExpandPosPerfectSquare

(a+b)2=a2+2ab+b2

y=3(x2+2x(1)+12)

BaseOne

1a=1

y=3(x2+2x(1)+1)

IdPropMult

Identity Property of Multiplication

y=3(x2+2x+1)

Distr

Distribute 3

y=3x2+6x+3

b A function graph is horizontally stretched or shrunk by multiplying the input of the function rule by a positive constant. If the constant is greater than 1, the graph is horizontally shrunk. Therefore, to horizontally shrink the graph by a factor of 2, the input of the function rule must be multiplied by 2.

Functiony=(x1)2HorizontalShrinkbyaFactorof2y=(2x1)2

Furthermore, a function graph is reflected in the x-axis by changing the sign of the output.

Functiony=(2x1)2Reflectioninthex-axisy=-(2x1)2

The described transformations are demonstrated by the following diagram.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (18)

The resulting function can be simplified by squaring the binomial and then distributing the -1.

y=-(2x1)2

Simplify right-hand side

ExpandNegPerfectSquare

(ab)2=a22ab+b2

y=-((2x)22(2x)(1)+12)

PowProdII

(ab)m=ambm

y=-(4x22(2x)(1)+12)

BaseOne

1a=1

y=-(4x22(2x)(1)+1)

Multiply

Multiply

y=-(4x24x+1)

Distr

Distribute -1

y=-4x2+4x1

Explore

Translating a Graph

In the coordinate plane, the graph of the quadratic function y=(xh)2+k can be seen. Observe how the graph is horizontally and vertically translated by changing the values of h and k.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (19)

Discussion

Translation of Quadratic Functions

A translation of a function is a transformation that shifts a graph vertically or horizontally. A vertical translation is achieved by adding some number to every output value of a function rule. Consider the quadratic function y=x2.

Functiony=x2VerticalTranslationbykUnitsy=x2+k

If k is a positive number, the translation is performed upwards. Conversely, if k is negative, the translation is performed downwards. If k=0, then there is no translation. This transformation can be shown on a coordinate plane.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (20)

A horizontal translation is instead achieved by subtracting a number from every input value.

Functiony=x2HorizontalTranslationbyhUnitsy=(xh)2

In this case, if h is a positive number, the translation is performed to the right. Conversely, if h is negative, the translation is performed to the left. If h=0, then there is no translation. It is worth noting that since h is subtracted from x, if h is positive, then a number is subtracted from x. On the other hand, if h is negative, a number is added to the variable x.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (21)

The vertical and horizontal translations of the graph of a function f can be summarized in a table.

Translation
Vertical Horizontal
Upwards
f(x)+k, with k>0
To the Right
f(xh), with h>0
Downwards
f(x)+k, with k<0
To the Left
f(xh), with h<0

Example

Translating a Quadratic Function

To finally finish the assignment and get the extra credit they need, Kriz has to finish the fourth task of the math assignment. This time, the graph of the quadratic parent function y=x2 is given.

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By translating this quadratic function, Kriz wants to draw the graphs and write the equations of the following functions.

a A translation of the graph of y=x2 three units up.

b A translation of the graph of y=x2 two units to the right.

c A translation of the graph of y=x2 one unit down and three units to the left.

Answer

a Equation: y=x2+3

Graph:

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (23)

b Equation: y=(x2)2

Graph:

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (24)

c Equation: y=(x+3)21

Graph:

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (25)

Hint

a The graph of y=f(x)+k is a vertical translation of the graph of y=f(x) by k units. If k is positive, the translation is upwards. If k is negative, the translation is downwards.

b The graph of y=f(xh) is a horizontal translation of the graph of y=f(x) by h units. If h is positive, the translation is to the right. If h is negative, the translation is to the left.

c The graph of y=f(xh)+k is a horizontal translation followed by a vertical translation by h and k units, respectively.

Solution

a The graph of y=f(x)+k is a vertical translation of the graph of y=f(x) by k units. If k is positive, the translation is upwards. Conversely, if k is negative, then the translation is downwards.

Functiony=x2Translation3UnitsUpy=x2+3

This can be seen on the coordinate plane.

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b The graph of y=f(xh) is a horizontal translation of the graph of y=f(x) by h units. If h is positive, the translation is to the right. Conversely, if h is negative, then the translation is to the left. Note that since h is subtracted from x, if h is positive, then the number is subtracted from x. On the other hand, if h is negative, the number is added to x.

Functiony=x2Translation2UnitstotheRighty=(x2)2

Here, since the given graph is to be translated 2 units to the right, the value of h is 2. Therefore, 2 is subtracted from the variable x.This can be seen on the coordinate plane.

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It is worth noting that the obtained function can be simplified by squaring the binomial.

y=(x2)2

Simplify right-hand side

ExpandNegPerfectSquare

(ab)2=a22ab+b2

y=x22x(2)+22

CommutativePropMult

Commutative Property of Multiplication

y=x22(2)x+22

CalcPow

Calculate power

y=x22(2)x+4

Multiply

Multiply

y=x24x+4

c The graph of y=f(xh)+k is a horizontal translation by h units followed by a vertical translation by k. If h is positive, the horizontal translation is to the right, and if it is negative, this translation is to the left. Similarly, if k is positive, the vertical translation is upwards. If k is negative, this translation is downwards.

Functiony=x2Translation1UnitDownand3UnitstotheLefty=(x+3)21

Again, special attention must be paid to the sign of h. This time, since the graph is to be translated 3 units to the left, the value of h is -3. Therefore, -3 must be subtracted from x. This is the same as adding 3 to x.

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Again, the obtained function can be simplified by squaring the binomial.

y=(x+3)21

Simplify right-hand side

ExpandPosPerfectSquare

(a+b)2=a2+2ab+b2

y=x2+2x(3)+321

CommutativePropMult

Commutative Property of Multiplication

y=x2+2(3)x+321

CalcPow

Calculate power

y=x2+2(3)x+91

Multiply

Multiply

y=x2+6x+91

SubTerm

Subtract term

y=x2+6x+8

Pop Quiz

Stating the Translation

The graph of the quadratic function y=x2 and a vertical or horizontal translation are shown in the coordinate plane.

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Closure

Transforming the Graph of a Function

With the topics learned in this lesson, the challenge presented at the beginning can be solved. The parabola that corresponds to the quadratic function f(x)=-x2+2x+1 is given.

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (30)

By applying transformations to the above graph, draw the graph of g(x)=-3x26x+1.

Answer

Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (31)

Hint

Rewrite the function whose graph is to be drawn to clearly identify the transformations.

Solution

First, the function will be rewritten to identify the transformations.

g(x)=-3x26x+1

Rewrite

WriteDiff

Write as a difference

g(x)=-3x26x+32

FactorOut

Factor out 3

g(x)=3(-x22x+1)2

NegBaseToPosBase

(-a)2=a2

g(x)=3(-(-x)22x+1)2

AddNeg

a+(-b)=ab

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

CommutativePropMult

Commutative Property of Multiplication

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

The expression in parenthesis corresponds to the right-hand side of the equation of f(x) but with a negative sign for the variable x. Therefore, g(x)=3f(-x)+(-2). According to the topics learned in this lesson, the graph of g can be explained as a sequence of transformations.

  1. Reflection in the y-axis.
  2. Vertical stretch by a factor of 3.
  3. Translation 2 units down.

This sequence of transformations can be seen on a coordinate plane.

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Mastering Parabola Transformations: Translating, Stretching, and Shrinking of Quadratic Functions (2024)
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