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.
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Challenge
Transforming a Function's Graph
The graph of the exponential function y=( 13)^x is drawn on the coordinate plane.
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Discussion
Exponential Parent Function
The exponential function y=b^x, 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.
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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=2^x. cc Function & Reflection in thex-axis y=2^x & y=- 2^x This reflection can be shown on a coordinate plane.
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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=0.5^x.
By reflecting this exponential parent function on the corresponding axis, she wants to draw the graphs of the following functions.
a y=- 0.5^x
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=- 0.5^x is a reflection of the graph of y=0.5^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. Therefore, the graph of y=0.5^(- x) is a reflection of the graph of y=0.5^x in the y-axis.
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Explore
Stretching and Shrinking a Graph
In the coordinate plane, the graph of the exponential function y=a(2^(cx)) can be seen. By changing the values of a and c, observe how the graph is vertically and horizontally stretched and shrunk.
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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=2^x. cc Function & Vertical Stretch/Shrink & by a Factor ofa y=2^x & y=a(2^x) 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.
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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=( 13)^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=( 13)^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=( 13)^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.
cc Function & Vertical Stretch & by a Factor of2 [0.8em] y=(1/3)^x & y=2(1/3)^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.
cc Function & Horizontal Shrink & by a Factor of5 [0.8em] y=(1/3)^x & y=(1/3)^(5x) This can be seen in a coordinate plane.
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Pop Quiz
Stating the Factor of a Stretch or a Shrink
The graph of the exponential parent function y=2^x 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 Exponential Functions
On her quest to figure out exponential functions, Tiffaniqua has met her match.
She is thinking about the exponential parent function y=4^x 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 12 followed by a reflection in the y-axis of the graph of y=4^x.
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b Next, she wants to write an exponential function whose graph is a horizontal stretch by a factor of 13 followed by a reflection in the x-axis of the graph of y=4^x.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"y=","formTextAfter":null,"answer":{"text":["-(4)^{\\dfrac{1}{3}x}","-4^{\\dfrac{1}{3}x}","-4^{\\dfrac{1}{3}\\t x}","\\N (4)^{\\dfrac{1}{3}\\t x}"]}}
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 12, the function rule must be multiplied by 12.
cc Function & Vertical Shrink & by a Factor of12 [0.8em] y=4^x & y=1/2(4)^x 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=1/2(4)^x & y=1/2(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 13, the input of the function rule must be multiplied by 13.
cc Function & Horizontal Stretch & by a Factor of13 [0.8em] y=4^x & y=4^(13x) Furthermore, a function graph is reflected in the x-axis by changing the sign of the y-coordinate of every point on the graph. cc Function & Reflection in thex-axis [0.8em] y=4^(13x) & y=- 4^(13x) This can be seen in a coordinate plane.
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Translating a Graph
In the coordinate plane, the graph of the function y=2^(x-h)+k can be seen. By changing the values of h and k, observe how the graph is horizontally and vertically translated.
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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=2^x. cc Function & Vertical Translation & bykUnits y=2^x & y=2^x+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.
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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=3^x.
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=3^x two units down.
b A translation of the graph of y=3^x one unit to the left.
c A translation of the graph of y=3^x two units up and three units to the right.
Answer
a Equation: y=3^x-2
Graph:
b Equation: y=3^(x+1)
Graph:
c Equation: y=3^(x-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.
Function y=3^x [0.5em] Translation2Units Down y=3^x+(- 2) ⇔ y=3^x-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.
Function y=3^x [0.5em] Translation1Unit to the Left y=3^(x-(- 1)) ⇔ y=3^(x+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.
Function y=3^x [0.5em] Translation2Units Up and3Units to the Right y=3^(x-3)+2 This can be seen on the coordinate plane.
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Pop Quiz
Stating the Translation
The graph of the exponential parent function y=2^x and a vertical or horizontal translation are shown in the coordinate plane.
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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=( 13)^x is given.
By applying transformations to the above graph, draw the graph of y= 12( 13)^(x+1)-2.
Answer
Hint
The graph of y= 12( 13)^(x+1)-2 is a translation 1 unit to the left, followed by a vertical shrink by a factor of 12, and a translation 2 units down of the graph of y=( 13)^x.
Solution
Consider the given function.
y= 12( 13)^(x+1)-2 ⇕ y= 12( 13)^(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 12, and a translation 2 units down of the graph of y=( 13)^x.
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Transformations of Exponential Functions
Exercises
Consider the following diagram.
The graph of g(x) is a vertical stretch of the graph of f(x). Find the stretch factor a.
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Looking at the given graph we notice that the y-intercept of g(x) is 2 and the y-intercept of f(x) is 1.
The y-intercept of g(x) can be obtained by multiplying the y-intercept of f(x) by 2. In fact, for the same input, the output of g(x) is twice the output of f(x). This means that we can obtain the graph of g(x) by stretching the graph of f(x) by a factor of 2.
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Solution
Consider the following function. f(x)=3^x Write a function g(x) whose graph is a horizontal translation 3 units to the left of the graph of f(x).
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["3^{x+3}"]}}
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Let's start by reviewing the form of a horizontal translation of an exponential function.
The exponential function g(x)=b^(x-h) is a horizontal translation of the parent function f(x) = b^x. If h is positive the function is translated h units to the right, and if h is negative the function is translated |h| units to the left.
To write a function that represents a horizontal translation of the parent function f(x) = 3^x, we just need to add or subtract from the input x. Since we want a translation 3 units to the left, we add 3 to the input. g(x)=f(x+ h) ⇕ g(x)=3^(x+ 3) To take a better look at the transformation, we will graph both functions on the same coordinate plane.
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Solution
Consider the following function. f(x)=2^x Write a function g(x) whose graph is a vertical translation 2 units down of the graph of f(x).
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["2^x-2"]}}
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Let's start by reviewing the form of a horizontal translation of an exponential function.
The exponential function g(x)=b^x+h is a vertical translation of the parent function f(x) = b^x. If h is positive the graph is translated h units up, and if h is negative the graph is translated |h| units down.
To write a function that represents a vertical translation of the parent function f(x) = 2^x we just need to add or subtract a number from the output of f(x). Since we want a translation 2 units down, we will subtract 2 from the output of f(x). g(x)=f(x)- 2 ⇕ g(x)=2^x- 2 To take a better look at the transformation, we will graph both functions on the same coordinate plane.
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Solution
Consider the following function. f(x)=3^x Write a function g(x) whose graph is a translation 2 units down and 1 unit to the right of the graph of f(x).
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"g(x)=","formTextAfter":null,"answer":{"text":["3^{x-1}-2"]}}
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We want to write a function for the indicated translation of the graph of f(x)=3^x. 2 units down and 1 unit to the right Before performing this translation, remember two important characteristics of translations.
- If a vertical translation is up, we add to the output. If the translation is down, we subtract from the output.
- If a horizontal translation is to the right, we subtract from the input. If the translation is to the left, we add to the input.
Given those characteristics, for the translation that we want to perform, we will have to subtract2 from the output of f(x), and subtract1 from its input. g(x)=f(x - 1) - 2 ⇕ g(x) = 3^(x - 1) - 2
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Solution
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{"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":["1.5^{-x}"]}}
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The graph of - f(x) is a reflection in the x-axis of the graph of f(x). This means that we will change the sign of the output of f(x). g(x) = - f(x) [0.5em] ⇕ [0.5em] g(x)=- 1.5^x
The graph of f(- x) is a reflection in the y-axis of the graph of f(x). This means that we will change the sign of the input of f(x).
h(x) = f(- x) [0.5em]
⇕ [0.5em]
h(x)=1.5^(- x)
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Solution
Transformations of Exponential Functions
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