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Graphing and Solving Exponential Equations and Inequalities

Transformations of Exponential Functions

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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.

Challenge

Transforming a Function's Graph

The graph of the exponential function is drawn on the coordinate plane.

exponential graph
By applying transformations to the above graph, draw the graph of

Discussion

Exponential Parent Function

The exponential function with b>0 and b1, 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.
b less than 1
Next, the graph of is shown for values of b grater than 1.
b greater 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
This reflection can be shown on a coordinate plane.
reflection on x-axis
A reflection in the y-axis is instead achieved by changing the sign of every input value.
This transformation can also be shown on a coordinate plane.
reflection on y-axis

Example

Reflecting an Exponential Function

Tiffaniqua is beginning to explore the graphs of exponential functions.

TiffaniquaReflections.svg

She is interested in the graph of the exponential function

parent function

By reflecting this exponential parent function on the corresponding axis, she wants to draw the graphs of the following functions.

a
b

Answer

a
reflection in the x-axis
b
reflection in the x-axis

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 is a reflection of the graph of in the x-axis.
parent function
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 is a reflection of the graph of in the y-axis.
parent function

Explore

Stretching and Shrinking a Graph

In the coordinate plane, the graph of the exponential function can be seen. By changing the values of a and c, observe how the graph is vertically and horizontally stretched and shrunk.

different values

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
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.
different values
Similarly, a function graph is horizontally stretched or shrunk by multiplying the input of a function rule by some positive constant c.
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.
different values

Example

Finding the Equation of a Stretch and a Shrink

Tiffaniqua is continuing her study of exponential functions.

ExponentialFunctions-Tiffaniqua.svg

She considers the exponential parent function and wants to write the function rules of two functions.

a An exponential function whose graph is a vertical stretch of the graph of by a factor of 2.
b An exponential function whose graph is a horizontal shrink of the graph of by a factor of 5.

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.
This can be seen in a coordinate plane.
vertical stretch
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.
This can be seen in a coordinate plane.
horizontal shrink

Pop Quiz

Stating the Factor of a Stretch or a Shrink

The graph of the exponential parent function 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 Exponential Functions

Tiffaniqua is stuck on a problem in her quest to figure out exponential functions.

TiffaniquaReflectionsHmm.svg

This time, she is thinking about the exponential parent function and wants to write the function rules of two functions.

a Write an exponential function whose graph is a vertical shrink by a factor of followed by a reflection in the y-axis of the graph of
b Write an exponential function whose graph is a horizontal stretch by a factor of followed by a reflection in the x-axis of the graph of

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 y-coordinate of every point on the graph.

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 the function rule must be multiplied by
Furthermore, a function graph is reflected in the y-axis by changing the sign of the input.
This can be seen in a coordinate plane.
combined transformations
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 the input of the function rule must be multiplied by
Furthermore, a function graph is reflected in the x-axis by changing the sign of the y-coordinate of every point on the graph.
This can be seen in a coordinate plane.
combined transformations

Explore

Translating a Graph

In the coordinate plane, the graph of the function can be seen. By changing the values of h and k, observe how the graph is horizontally and vertically translated.

different values

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
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.
different values
A horizontal translation is instead achieved by subtracting a number from every input value.
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.
horizontal translation

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.

TiffaniquaTranslations.svg

Tiffaniqua has drawn the graph of the exponential function

exponential function

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 two units down.
b A translation of the graph of one unit to the left.
c A translation of the graph of two units up and three units to the right.

Answer

a Equation:

Graph:

translation 2 units down
b Equation:

Graph:

translation 1 unit to the left
c Equation:

Graph:

translation 2 units up 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(xh) 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(xh)+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.
This can be seen on the coordinate plane.
translation down
b The graph of y=f(xh) 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.
This can be seen on the coordinate plane.
translation to the left
c The graph of y=f(xh)+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.
This can be seen on the coordinate plane.
combined translations

Pop Quiz

Stating the Translation

The graph of the exponential parent function and a vertical or horizontal translation are shown in the coordinate plane.

find the value of the constant

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 is given.

exponential graph
By applying transformations to the above graph, draw the graph of

Answer

exponential graph

Hint

The graph of is a translation 1 unit to the left, followed by a vertical shrink by a factor of and a translation 2 units down of the graph of

Solution

Consider the given function.
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 and a translation 2 units down of the graph of
exponential graph
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