4. Transformations of Quadratic Functions
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Chapter 9
4.
Transformations of Quadratic Functions
This lesson delves into the fascinating world of quadratic functions, specifically focusing on how these functions can be transformed. It covers horizontal and vertical translations, stretches, and shrinks. For example, if you're an architect designing a parabolic arch, understanding these transformations can help you adjust the curve to fit specific dimensions. Similarly, in physics, these transformations can be used to model the trajectory of a projectile. The lesson also provides practical examples, like a student named Kriz who is tasked with various transformations of quadratic functions. It even discusses more complex transformations that combine multiple types of shifts and stretches.
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Lesson Settings & Tools
| | 14 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Transformations of Quadratic Functions
Slide of 14
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.
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Challenge
Transforming the Graph of a Function
The graph of the quadratic function f(x)=- x^2+2x+1 is drawn on the coordinate plane. By applying transformations to the parabola that corresponds to f, draw the graph of g(x)=- x^2+4x.
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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=x^2. cc Function & Reflection in thex-axis y=x^2 & y=- x^2 The reflection of the corresponding parabola can 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= 12(x+2)^2+1.
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:
Equation: y=- 1/2(x+2)^2-1
b Graph:
Equation: y=1/2(x-2)^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).
Given Function f(x)=1/2(x+2)^2+1 [1em] Reflection in thex -axis - f(x)=- (1/2(x+2)^2+1) ⇕ - f(x)=- 1/2(x+2)^2-1 Therefore, the graph of y=- 12(x+2)^2-1 is a reflection of the graph of y= 12(x+2)^2+1 in the x-axis.
b The reflection of the graph of y=f(x) in the y-axis is given by the equation of y=f(- x).
Given Function f(x)=1/2(x+2)^2+1 [1em] Reflection in they -axis f(- x)=1/2(- x+2)^2+1 To obtained equation can be simplified.
f(- x)=1/2(- x+2)^2+1
NegBaseToPosPow
(- a)^2 = a^2
f(- x)=1/2(- (- x+2))^2+1
Distr
Distribute - 1
f(- x)=1/2(x-2)^2+1
Therefore, the graph of y= 12(x-2)^2+1 is a reflection of the graph of y= 12(x+2)^2+1 in the y-axis.
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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=x^2+1. cc Function & Vertical Stretch/Shrink & by a Factor ofa y=x^2+1 & y=a(x^2+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.
| Vertical | Horizontal | |
|---|---|---|
| Stretch | af(x), with a>1 | f(ax), with 0 |
| Shrink | af(x), with 0 | f(ax), with a>1 |
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Example
Finding the Equation of a Stretch and a Shrink
Kriz's second task is about horizontal and vertical stretches and shrinks of parabolas.
They are given the quadratic function y=x^2-3 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=x^2-3 by a factor of 2.
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b A quadratic function whose graph is a horizontal shrink of the graph of y=x^2-3 by a factor of 4.
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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.
cc Function & Vertical Stretch & by a Factor of2 [0.8em] y=x^2-3 & y=2(x^2-3) 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.
It is worth noting that the resulting function can be simplified by distributing 2. y=2(x^2-3) ⇔ y=2x^2-6
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.
cc Function & Horizontal Shrink & by a Factor of4 [0.8em] y=x^2-3 & y=(4x)^2-3 This can be seen in a coordinate plane.
It is worth noting that the resulting function can be simplified by using the Power of a Product Property. y=(4x)^2-3 ⇔ y=16x^2-3
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Pop Quiz
Stating the Factor of a Stretch or a Shrink
The graph of the quadratic function y=x^2 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.
This time, they are given the quadratic function y=(x-1)^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=(x-1)^2.
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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=(x-1)^2.
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Hint
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.
cc Function & Vertical Stretch & by a Factor of3 [0.8em] y=(x-1)^2 & y= 3(x-1)^2 Furthermore, a function graph is reflected in the y-axis by changing the sign of the input. cc Function & Reflection in they-axis [0.8em] y=3(x-1)^2 & y=3( - x-1)^2 These transformations are illustrated by the following diagram.
The resulting function can be simplified by squaring the binomial and then distributing the 3.
y=3(- x-1)^2
▼
Simplify right-hand side
NegBaseToPosPow
(- a)^2 = a^2
y=3(-(- x-1))^2
Distr
Distribute -1
y=3(x+1)^2
ExpandPosPerfectSquare
(a+b)^2=a^2+2ab+b^2
y=3(x^2+2x(1)+1^2)
BaseOne
1^a=1
y=3(x^2+2x(1)+1)
IdPropMult
Identity Property of Multiplication
y=3(x^2+2x+1)
Distr
Distribute 3
y=3x^2+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.
cc Function & Horizontal Shrink & by a Factor of2 [0.8em] y=(x-1)^2 & y=( 2x-1)^2 Furthermore, a function graph is reflected in the x-axis by changing the sign of the output. cc Function & Reflection in thex-axis [0.8em] y=(2x-1)^2 & y= - (2x-1)^2 The described transformations are demonstrated by the following diagram.
The resulting function can be simplified by squaring the binomial and then distributing the - 1.
y=- (2x-1)^2
▼
Simplify right-hand side
ExpandNegPerfectSquare
(a-b)^2=a^2-2ab+b^2
y=- ((2x)^2-2(2x)(1)+1^2)
PowProdII
(a b)^m=a^m b^m
y=- (4x^2-2(2x)(1)+1^2)
BaseOne
1^a=1
y=- (4x^2-2(2x)(1)+1)
Multiply
Multiply
y=- (4x^2-4x+1)
Distr
Distribute - 1
y=- 4x^2+4x-1
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Translating a Graph
In the coordinate plane, the graph of the quadratic function y=(x-h)^2+k can be seen. Observe how the graph is horizontally and vertically translated by changing the values of h and k.
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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=x^2. cc Function & Vertical Translation & bykUnits y=x^2 & y=x^2+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.
| Translation | |
|---|---|
| Vertical | Horizontal |
| Upwards f(x)+k, with k>0 |
To the Right f(x-h), with h>0 |
| Downwards f(x)+k, with k<0 |
To the Left f(x-h), with h<0 |
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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=x^2 is given.
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=x^2 three units up.
b A translation of the graph of y=x^2 two units to the right.
c A translation of the graph of y=x^2 one unit down and three units to the left.
Answer
a Equation: y=x^2+3
Graph:
b Equation: y=(x-2)^2
Graph:
c Equation: y=(x+3)^2-1
Graph:
Hint
b The graph of y=f(x-h) 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(x-h)+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.
cc Function & Translation3Units Up y=x^2 & y=x^2+3 This can be seen on the coordinate plane.
b The graph of y=f(x-h) 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.
cc Function & Translation2Units to the Right y=x^2 & y=(x-2)^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.
It is worth noting that the obtained function can be simplified by squaring the binomial.
y=(x-2)^2
▼
Simplify right-hand side
ExpandNegPerfectSquare
(a-b)^2=a^2-2ab+b^2
y=x^2-2x(2)+2^2
CommutativePropMult
Commutative Property of Multiplication
y=x^2-2(2)x+2^2
CalcPow
Calculate power
y=x^2-2(2)x+4
Multiply
Multiply
y=x^2-4x+4
c The graph of y=f(x-h)+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.
cc Function & Translation1Unit Down & and3Units to the Left y=x^2 & y=(x+3)^2-1 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.
Again, the obtained function can be simplified by squaring the binomial.
y=(x+3)^2-1
▼
Simplify right-hand side
ExpandPosPerfectSquare
(a+b)^2=a^2+2ab+b^2
y=x^2+2x(3)+3^2-1
CommutativePropMult
Commutative Property of Multiplication
y=x^2+2(3)x+3^2-1
CalcPow
Calculate power
y=x^2+2(3)x+9-1
Multiply
Multiply
y=x^2+6x+9-1
SubTerm
Subtract term
y=x^2+6x+8
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Stating the Translation
The graph of the quadratic function y=x^2 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)=- x^2+2x+1 is given.
By applying transformations to the above graph, draw the graph of g(x)=- 3x^2-6x+1.
Answer
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)=- 3x^2-6x+1
▼
Rewrite
WriteDiff
Write as a difference
g(x)=- 3x^2-6x+3-2
FactorOut
Factor out 3
g(x)=3(- x^2-2x+1)-2
NegBaseToPosBase
(- a)^2=a^2
g(x)=3(- (- x)^2-2x+1)-2
AddNeg
a+(- b)=a-b
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.
- Reflection in the y-axis.
- Vertical stretch by a factor of 3.
- Translation 2 units down.
This sequence of transformations can be seen on a coordinate plane.
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Transformations of Quadratic Functions
Exercise 2.1
Two quadratic functions are graphed on the same coordinate plane.
Find the equation of h(x). Write your answer in standard form.
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We can see in the given diagram that the graph of h(x) is a translation 5 units to the left of the graph of f(x).
To translate the graph of a function 5 units to the left, we add 5 to the function's input. This means that we need to calculate an expression for f(x+5). h(x)=f(x+5) ⇕ h(x)=(x+5)^2+(x+5)-1 Let's simplify the right-hand side of this equation.
To check our answer, we can draw the graph of h(x)=x^2+11x+29 and verify that this parabola matches the given graph. To do so, we will first make a table of values.
| x | x^2+11x+29 | h(x) |
|---|---|---|
| - 7 | ( - 7)^2+11( - 7)+29 | 1 |
| - 6 | ( - 6)^2+11( - 6)+29 | - 1 |
| - 5 | ( - 5)^2+11( - 5)+29 | - 1 |
| - 4 | ( - 4)^2+11( - 4)+29 | 1 |
By using this table, we obtained the points (- 7,1), (- 6,- 1), (- 5,- 1), and (- 4,1). Let's plot and connect them with a smooth curve. We will also include the graph of f(x)=x^2+x-1 on the same coordinate plane to see their relationship.
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Solution
Two quadratic functions are graphed on the same coordinate plane.
Find the equation of h(x).
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We can see in the given diagram that the graph of h(x) is wider than the graph of f(x). We can also see that both parabolas have the same vertex. These things suggest that the graph of h(x) is a horizontal stretch of the graph of f(x).
What is more, by counting the units, we can see that the graph of h(x) is four times wider than the graph of f(x). Therefore, the graph of h(x) is a horizontal stretch of the graph of f(x) by a factor of 14. To horizontally stretch the graph of a function by a factor of 14, we multiply the function's input by 14. This means that we need to find an expression for f( 14x). h(x)=f( 14x) ⇕ h(x)=(1/4x)^2-1 Let's simplify the right-hand side of the above equation by using properties of exponents.
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Solution
Two quadratic functions are graphed on the same coordinate plane.
Find the equation of h(x).
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"h(x)=","formTextAfter":null,"answer":{"text":["-x^2+4x-1","-1(x-2)^2+3","-(x-2)^2+3"]}}
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We can see in the given diagram that the x-axis acts like a mirror. Therefore, the graph of h(x) is a reflection of the graph of f(x) in the x-axis.
To reflect the graph of a function in the x-axis, we multiply the function rule by - 1. This means that we need to find an expression for - f(x). h(x)=- f(x) ⇕ h(x)=- (x^2-4x+1) We can simplify the above equation by distributing - 1 on the right-hand side.
To check our answer, we can draw the graph of h(x)=- x^2+4x-1 and verify that this parabola matches the given graph. To do so, we will first make a table of values.
| x | - x^2+4x-1 | h(x) |
|---|---|---|
| 0 | - 0^2+4( 0)-1 | - 1 |
| 1 | - 1^2+4( 1)-1 | 2 |
| 2 | - 2^2+4( 2)-1 | 3 |
| 3 | - 3^2+4( 3)-1 | 2 |
| 4 | - 4^2+4( 4)-1 | - 1 |
By using this table, we obtained the points (0,- 1), (1,2), (2,3), (3,2), and (4,- 1). Let's plot and connect them with a smooth curve. We will also include the graph of f(x)=x^2-4x+1 on the same coordinate plane to see their relationship.
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Solution
Transformations of Quadratic Functions
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