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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.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

The graph of the exponential function $y=(31 )_{x}$ is drawn on the coordinate plane.

By applying transformations to the above graph, draw the graph of $y=21 (31 )_{x+1}−2.$Discussion

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 $1.$

Next, the graph of $y=b_{x}$ is shown for values of $b$ grater than $1.$

Discussion

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 $y=2_{x}.$

$Functiony=2_{x} Reflection in thex-axisy=-2_{x} $

This reflection can be shown on a coordinate plane.
A reflection in the $y-$axis is instead achieved by changing the sign of every input value.

$Functiony=2_{x} Reflection in they-axisy=2_{-x} $

This transformation can also be shown on a coordinate plane. Example

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 $y=0.5_{-x}$

a

b

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.

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.

Explore

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.

Discussion

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 $y=2_{x}.$

$Functiony=2_{x} Vertical Stretch/Shrinkby a Factor ofay=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 $a.$
Similarly, a function graph is horizontally stretched or shrunk by multiplying the input of a function rule by some positive constant $c.$

$Functiony=2_{x} Horizontal Stretch/Shrinkby a Factor ofcy=2_{cx} $

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

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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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 )_{x} Vertical Stretchby a Factor of2y=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 )_{x} Horizontal Shrinkby a Factor of5y=(31 )_{5x} $

This can be seen in a coordinate plane.
Pop Quiz

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.

Example

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 $21 $ 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 $31 $ followed by a reflection in the $x-$axis of the graph of $y=4_{x}.$

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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=4_{x} Vertical Shrinkby a Factor of21 y=21 (4)_{x} $

Furthermore, a function graph is reflected in the $y-$axis by changing the sign of the input.
$Functiony=21 (4)_{x} Reflection in they-axisy=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=4_{x} Horizontal Stretchby a Factor of31 y=4_{31x} $

Furthermore, a function graph is reflected in the $x-$axis by changing the sign of the $y-$coordinate of every point on the graph.
$Functiony=4_{31x} Reflection in thex-axisy=-4_{31x} $

This can be seen in a coordinate plane.
Explore

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.

Discussion

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 $y=2_{x}.$

$Functiony=2_{x} Vertical TranslationbykUnitsy=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.
A horizontal translation is instead achieved by subtracting a number from every input value.

$Functiony=2_{x} Horizontal TranslationbyhUnitsy=2_{x−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

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.

a **Equation:** $y=3_{x}−2$

**Graph:**

b **Equation:** $y=3_{x+1}$

**Graph:**

c **Equation:** $y=3_{x−3}+2$

**Graph:**

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.

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=3_{x}Translation2Units Downy=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.

$Functiony=3_{x}Translation1Unit to the Lefty=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.

$Functiony=3_{x}Translation2Units Upand3Units to the Righty=3_{x−3}+2 $

This can be seen on the coordinate plane.
Pop Quiz

The graph of the exponential parent function $y=2_{x}$ and a vertical or horizontal translation are shown in the coordinate plane.

Closure

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.

By applying transformations to the above graph, draw the graph of $y=21 (31 )_{x+1}−2.$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 $y=(31 )_{x}.$

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 $y=(31 )_{x}.$
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