4. Transformations of Exponential Functions
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Chapter 6
4.
Transformations of Exponential Functions
Tiffaniqua embarks on a journey to understand the graphs of exponential functions. She explores various transformations, including vertical and horizontal translations, stretches, shrinks, and reflections. Through her exploration, she learns how to identify and apply these transformations to different exponential functions. These transformations play a crucial role in understanding the behavior of exponential graphs and their real-world implications. By mastering these concepts, one can effectively interpret and utilize exponential functions in various scenarios.
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Lesson Settings & Tools
| | 15 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Transformations of Exponential Functions
Slide of 15
This lesson will discuss how to apply different transformations to exponential functions. Also, it will be seen how to identify these transformations in the graphs of different exponential functions.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Transforming a Function's Graph
The graph of the exponential function y=(31)x is drawn on the coordinate plane.
Discussion
Exponential Parent Function
The exponential function y=bx, with b>0 and b=1, is the exponential parent function. This is the most basic function of the family. The graph of this function is shown for positive values of b less than 1.
Next, the graph of y=bx is shown for values of b grater than 1. 
Discussion
Reflection of Exponential 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. This means changing the sign of the y-coordinate of every point on the graph of a function. Consider the exponential parent function y=2x.
A reflection in the y-axis is instead achieved by changing the sign of every input value.
Functiony=2xReflection in the x-axisy=-2x
This reflection can be shown on a coordinate plane.
Functiony=2xReflection in the y-axisy=2-x
This transformation can also be shown on a coordinate plane. Example
Reflecting an Exponential Function
Tiffaniqua is beginning to explore the graphs of exponential functions.
She is interested in the graph of the exponential function y=0.5x.
By reflecting this exponential parent function on the corresponding axis, she wants to draw the graphs of the following functions.
a y=-0.5x
b y=0.5-x
Answer
a
b
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 graph of y=-f(x) is a reflection of the graph of y=f(x) in the x-axis. Therefore, the graph of y=-0.5x is a reflection of the graph of y=0.5x 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. Therefore, the graph of y=0.5-x is a reflection of the graph of y=0.5x in the y-axis.
Explore
Stretching and Shrinking a Graph
In the coordinate plane, the graph of the exponential function y=a(2cx) can be seen. By changing the values of a and c, observe how the graph is vertically and horizontally stretched and shrunk.
Discussion
Stretch and Shrink of Exponential Functions
A function graph is vertically stretched or shrunk by multiplying the output of a function rule by some positive constant a. Consider the exponential parent function y=2x.
Similarly, a function graph is horizontally stretched or shrunk by multiplying the input of a function rule by some positive constant c.
Functiony=2xVertical Stretch/Shrinkby a Factor of ay=a(2x)
If a is greater than 1, the graph 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. All vertical distances from the graph to the x-axis are changed by the factor a.
Functiony=2xHorizontal Stretch/Shrinkby a Factor of cy=2cx
In this case, if c is greater than 1, the graph is horizontally shrunk by a factor of c. Conversely, if c is less than 1, the graph is horizontally stretched by a factor of c. If c=1, then there is neither a stretch nor shrink of the graph. Example
Finding the Equation of a Stretch and a Shrink
Tiffaniqua, feeling good, continues her study of exponential functions.
She considers the exponential parent function y=(31)x and wants to write the function rules of two functions.
a An exponential function whose graph is a vertical stretch of the graph of y=(31)x by a factor of 2.
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b An exponential function whose graph is a horizontal shrink of the graph of y=(31)x by a factor of 5.
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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.
Functiony=(31)xVertical Stretchby a Factor of 2y=2(31)x
This can be seen in a coordinate plane.
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 5, the input of the function rule must be multiplied by 5.
Functiony=(31)xHorizontal Shrinkby a Factor of 5y=(31)5x
This can be seen in a coordinate plane.
Pop Quiz
Stating the Factor of a Stretch or a Shrink
The graph of the exponential parent function y=2x is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.
Example
Combining Transformations of Exponential Functions
On her quest to figure out exponential functions, Tiffaniqua has met her match.
She is thinking about the exponential parent function y=4x and wants to write the function rules of two functions.
a First, she wants write an exponential function whose graph is a vertical shrink by a factor of 21 followed by a reflection in the y-axis of the graph of y=4x.
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b Next, she wants to write an exponential function whose graph is a horizontal stretch by a factor of 31 followed by a reflection in the x-axis of the graph of y=4x.
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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 less than 1, the graph is vertically shrunk. Therefore, if the graph of the given function is to be vertically shrunk by a factor of 21, the function rule must be multiplied by 21.
Functiony=4xVertical Shrinkby a Factor of 21y=21(4)x
Furthermore, a function graph is reflected in the y-axis by changing the sign of the input.
Function y=21(4)x Reflection in the y-axis y=21(4)-x
This can be seen in a coordinate plane.
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 less than 1, the graph is horizontally stretched. Therefore, to horizontally stretch the graph by a factor of 31, the input of the function rule must be multiplied by 31.
Functiony=4xHorizontal Stretchby a Factor of 31y=431x
Furthermore, a function graph is reflected in the x-axis by changing the sign of the y-coordinate of every point on the graph.
Function y=431x Reflection in the x-axis y=-431x
This can be seen in a coordinate plane.
Explore
Translating a Graph
In the coordinate plane, the graph of the function y=2x−h+k can be seen. By changing the values of h and k, observe how the graph is horizontally and vertically translated.
Discussion
Translation of Exponential 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 exponential parent function y=2x.
A horizontal translation is instead achieved by subtracting a number from every input value.
Functiony=2xVertical Translationby k Unitsy=2x+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.
Functiony=2xHorizontal Translationby h Unitsy=2x−h
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. This transformation can also be shown on a coordinate plane. Example
Translating an Exponential Function
Tiffaniqua is feeling good about her understanding of the graphs of exponential functions. She is now moving on to mastering another aspect of them.
With translations on her mind, Tiffaniqua has drawn the graph of the exponential function y=3x.
By translating this exponential parent function, she wants to draw the graphs and write the equations of the following functions.
a A translation of the graph of y=3x two units down.
b A translation of the graph of y=3x one unit to the left.
c A translation of the graph of y=3x two units up and three units to the right.
Answer
a Equation: y=3x−2
Graph:
b Equation: y=3x+1
Graph:
c Equation: y=3x−3+2
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.
Functiony=3xTranslation 2 Units Downy=3x+(-2)⇔y=3x−2
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.
Functiony=3xTranslation 1 Unit to the Lefty=3x−(-1)⇔y=3x+1
This can be seen on the coordinate plane.
c The graph of y=f(x−h)+k is a horizontal translation followed by a vertical translation by h and k units, respectively. If h is positive the horizontal translation is to the right, and if h 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=3xTranslation 2 Units Upand 3 Units to the Righty=3x−3+2
This can be seen on the coordinate plane.
Pop Quiz
Stating the Translation
The graph of the exponential parent function y=2x and a vertical or horizontal translation are shown in the coordinate plane.
Closure
Transforming a Function's Graph
With the topics learned in this lesson, the challenge presented at the beginning can be solved. The graph of the exponential function y=(31)x is given.
Answer
Hint
The graph of y=21(31)x+1−2 is a translation 1 unit to the left, followed by a vertical shrink by a factor of 21, and a translation 2 units down of the graph of y=(31)x.
Solution
Consider the given function.
y=21(31)x+1−2⇕y=21(31)x−(-1)+(-2)
According to the topics learned in this lesson, the graph of the above function is a translation 1 unit to the left, followed by a vertical shrink by a factor of 21, and a translation 2 units down of the graph of y=(31)x.
Transformations of Exponential Functions
Exercise 2.1
Marcos is in the lab of his biology class. The students are given a microscope to take a look at a population of bacteria.
p(x)=10⋅2x
Marcos knows that the initial population of bacteria is 10. He wants to do a new experiment with twice the initial amount of bacteria. The function that models this new experiment is labeled r(x). What is the only statement that is true?
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Marcos wants to conduct a new experiment using twice the initial amount of bacteria. In the given exponential function, the initial population is 10. Therefore, to double the initial amount, we will multiply 10 by 2. r(x)=( 2* 10)* 2^x ⇕ r(x)=20* 2^x By using the Associative Property of Multiplication, the resulting function can be written as 2 times p(x).
This means that the graph of r(x) is a vertical stretch of the graph of p(x) by a factor of 2.
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Mark has tried to translate an exponential function 2 units to the right and 1 unit up. He followed the steps shown below. However, his teacher told him that the answer is incorrect.
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Mark wants to write an exponential function for the indicated translation of the graph of f(x)=2* 5^x. 1 unit up and 2 units to the right In order to find Mark's mistake, we will begin by recalling two things.
- The graph of f(x)+k is a vertical translation of the graph of f(x) by k units. If k is positive, the translation is upwards. If k<0, the translation is downwards.
- The graph of f(x-h) is a horizontal translation of the graph of f(x) by h units. If h is positive, the translation is to the right. If h<0, the translation is to the left.
Therefore, for the indicated translation we have to add1 to the output of f(x), and subtract2 from its input. f(x - 2) + 1 = 2* 5^(x - 2) + 1 We notice that Mark added 2 to the input instead of subtracting. This is the mistake. Conversely, the second step is correctly done. Therefore, only step 1 is wrong.
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Transformations of Exponential Functions
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