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.
- Graphing Quadratic Functions in Standard Form
- Graphing Quadratic Functions in Vertex Form and Intercept Form
- Translation of a Function
- Vertical Stretch and Shrink of a Function
- Horizontal Stretch and Shrink of a Function
- Reflection of a Function
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.
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.
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=(x−1)2Reflectioninthey-axisy=(-x−1)2
This transformation can also be shown on a coordinate plane.
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.
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=-21(x+2)2−1
b Graph:
Equation: y=21(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).
GivenFunctionf(x)=21(x+2)2+1Reflectioninthex-axis-f(x)=-(21(x+2)2+1)⇕-f(x)=-21(x+2)2−1
Therefore, the graph of y=-21(x+2)2−1 is a reflection of the graph of y=21(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).
GivenFunctionf(x)=21(x+2)2+1Reflectioninthey-axisf(-x)=21(-x+2)2+1
To obtained equation can be simplified.
Therefore, the graph of y=21(x−2)2+1 is a reflection of the graph of y=21(x+2)2+1 in the y-axis.
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.
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.
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.
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 |
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=x2−3 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=x2−3 by a factor of 2.
b A quadratic function whose graph is a horizontal shrink of the graph of y=x2−3 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=x2−3VerticalStretchbyaFactorof2y=2(x2−3)
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(x2−3)⇔y=2x2−6
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.
Functiony=x2−3HorizontalShrinkbyaFactorof4y=(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=16x2−3
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.
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.
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.
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=(x−1)2VerticalStretchbyaFactorof3y=3(x−1)2
Furthermore, a function graph is reflected in the y-axis by changing the sign of the input.
Functiony=3(x−1)2Reflectioninthey-axisy=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
(-a)2=a2
y=3(-(-x−1))2
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)
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=(x−1)2HorizontalShrinkbyaFactorof2y=(2x−1)2
Furthermore, a function graph is reflected in the x-axis by changing the sign of the output.
Functiony=(2x−1)2Reflectioninthex-axisy=-(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
(a−b)2=a2−2ab+b2
y=-((2x)2−2(2x)(1)+12)
PowProdII
(ab)m=ambm
y=-(4x2−2(2x)(1)+12)
BaseOne
1a=1
y=-(4x2−2(2x)(1)+1)
Multiply
y=-(4x2−4x+1)
Distribute -1
y=-4x2+4x−1
Explore
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.
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.
A horizontal translation is instead achieved by subtracting a number from every input value.
Functiony=x2HorizontalTranslationbyhUnitsy=(x−h)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.
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(x−h), with h>0 |
| Downwards f(x)+k, with k<0 | To the Left f(x−h), 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.
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:
b Equation: y=(x−2)2
Graph:
c Equation: y=(x+3)2−1
Graph:
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(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.
Functiony=x2Translation3UnitsUpy=x2+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.
Functiony=x2Translation2UnitstotheRighty=(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
(a−b)2=a2−2ab+b2
y=x2−2x(2)+22
Commutative Property of Multiplication
y=x2−2(2)x+22
CalcPow
Calculate power
y=x2−2(2)x+4
Multiply
y=x2−4x+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.
Functiony=x2Translation1UnitDownand3UnitstotheLefty=(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=a2+2ab+b2
y=x2+2x(3)+32−1
Commutative Property of Multiplication
y=x2+2(3)x+32−1
CalcPow
Calculate power
y=x2+2(3)x+9−1
Multiply
y=x2+6x+9−1
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.
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.
By applying transformations to the above graph, draw the graph of g(x)=-3x2−6x+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)=-3x2−6x+1
▼
Rewrite
WriteDiff
Write as a difference
g(x)=-3x2−6x+3−2
FactorOut
Factor out 3
g(x)=3(-x2−2x+1)−2
NegBaseToPosBase
(-a)2=a2
g(x)=3(-(-x)2−2x+1)−2
AddNeg
a+(-b)=a−b
g(x)=3(-(-x)2+(-2x)+1)+(-2)
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.
