A1
Algebra 1 View details
4. Transformations of Quadratic Functions
Continue to next lesson
Lesson
Exercises
Tests
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.
Show more expand_more
Problem Solving Reasoning and Communication Error Analysis Modeling Using Tools Precision Pattern Recognition
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
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.

Parabola
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.
Reflection of the parabola in x-axis
A reflection in the y-axis is instead achieved by changing the sign of every input value. However, since (- x)^2 is equivalent to x^2, reflecting y=x^2 in the y-axis does not change the graph. For this reason, the reflection of the graph of another quadratic function will be shown. cc Function & Reflection in they-axis y=(x-1)^2 & y=(- x-1)^2 This transformation can also be shown on a coordinate plane.
Reflection of y=(x-1)^2 in y-axis
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.

Parabola of the given function

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:
reflection in the x-axis

Equation: y=- 1/2(x+2)^2-1

b Graph:
reflection in the y-axis

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.
The given function is being reflected over 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
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 y-axis.
The given function is being reflected over 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.

Illustration of how changes of the coefficients of a and b affect the graph of the parabola
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.
Different values of the coefficient a lead to the graph stretching or shrinking vertically
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. cc Function & Horizontal Stretch/Shrink & by a Factor ofb y=x^2+1 & y=(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 1b.
Different values of the coefficient b lead to the graph stretching or shrinking horizontally
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
Shrink af(x), with 0 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.

Kriz studying

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.
b A quadratic function whose graph is a horizontal shrink of the graph of y=x^2-3 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.
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.
vertical stretch
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.
horizontal shrink
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
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.

Find the value of the constant
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.

Kriz studying

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.
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.
Combined transformations of y=(x-1)^2
The resulting function can be simplified by squaring the binomial and then distributing the 3.
y=3(- x-1)^2
Simplify right-hand side
y=3(-(- x-1))^2
y=3(x+1)^2
y=3(x^2+2x(1)+1^2)
y=3(x^2+2x(1)+1)
y=3(x^2+2x+1)
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.
Combined transformations
The resulting function can be simplified by squaring the binomial and then distributing the - 1.
y=- (2x-1)^2
Simplify right-hand side
y=- ((2x)^2-2(2x)(1)+1^2)
y=- (4x^2-2(2x)(1)+1^2)
y=- (4x^2-2(2x)(1)+1)
y=- (4x^2-4x+1)
y=- 4x^2+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.

Changing values of h and k affects the graph of the function
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.
Translating a parabola vertically
A horizontal translation is instead achieved by subtracting a number from every input value. cc Function & Horizontal Translation & byhUnits y=x^2 & y=(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.
Translating a parabola horizontally
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=x^2 is given.

quadratic function

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:

translation 3 units up
b Equation: y=(x-2)^2

Graph:

translation 2 units to the right
c Equation: y=(x+3)^2-1

Graph:

translation 1 unit down and 3 units to the left

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.
cc Function & Translation3Units Up y=x^2 & y=x^2+3 This can be seen on the coordinate plane.
translation down
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.
translation to the left
It is worth noting that the obtained function can be simplified by squaring the binomial.
y=(x-2)^2
Simplify right-hand side
y=x^2-2x(2)+2^2
y=x^2-2(2)x+2^2
y=x^2-2(2)x+4
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.
combined translations
Again, the obtained function can be simplified by squaring the binomial.
y=(x+3)^2-1
Simplify right-hand side
y=x^2+2x(3)+3^2-1
y=x^2+2(3)x+3^2-1
y=x^2+2(3)x+9-1
y=x^2+6x+9-1
y=x^2+6x+8
Pop Quiz

Stating the Translation

The graph of the quadratic function y=x^2 and a vertical or horizontal translation are shown in the coordinate plane.

Find the value of the constant
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.

parabola
By applying transformations to the above graph, draw the graph of g(x)=- 3x^2-6x+1.

Answer

transformed quadratic graph

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
g(x)=- 3x^2-6x+3-2
g(x)=3(- x^2-2x+1)-2
g(x)=3(- (- x)^2-2x+1)-2
g(x)=3(- (- x)^2+(- 2x)+1)+(- 2)
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.
quadratic graph
Transformations of Quadratic Functions
Exercises
>
2
e
7
8
9
×
÷1
=
=
4
5
6
+
<
log
ln
log
1
2
3
()
sin
cos
tan
0
.
π
x
y