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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.

Graphing exponential functions
Translation of a function
Vertical stretch and shrink of a function
Horizontal stretch and shrink of a function
Reflection of a function

Challenge

Transforming a Function's Graph

The graph of the exponential function $y = (\frac{1}{3})^{x}$ is drawn on the coordinate plane.

exponential graph

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

y = \frac{1}{2} (\frac{1}{3})^{x + 1} - 2 .

Discussion

Exponential Parent Function

The exponential function

y = b^{x},

with

b > 0

and

b \neq = 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 .

b less than 1

Next, the graph of

y = b^{x}

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

y = 2^{x} .

Function y = 2^{x} Reflection in the x - axis y = - 2^{x}

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.

Function y = 2^{x} Reflection in the y - axis y = 2^{- x}

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.

She is interested in the graph of the exponential function $y = 0 . 5^{x} .$

parent function

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.

Explore

Stretching and Shrinking a Graph

In the coordinate plane, the graph of the exponential function $y = a (2^{c x})$ 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

y = 2^{x} .

Function y = 2^{x} Vertical Stretch / Shrink by a Factor of a 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 .

different values

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

c .

Function y = 2^{x} Horizontal Stretch / Shrink by a Factor of c y = 2^{c x}

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, feeling good, continues her study of exponential functions.

She considers the exponential parent function $y = (\frac{1}{3})^{x}$ 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 = (\frac{1}{3})^{x}

by a factor of

2 .

b An exponential function whose graph is a horizontal shrink of the graph of

y = (\frac{1}{3})^{x}

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 .

Function y = (\frac{1}{3})^{x} Vertical Stretch by a Factor of 2 y = 2 (\frac{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 .

Function y = (\frac{1}{3})^{x} Horizontal Shrink by a Factor of 5 y = (\frac{1}{3})^{5 x}

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 $y = 2^{x}$ 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

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

\frac{1}{2}

followed by a reflection in the

y -

axis of the graph of

y = 4^{x} .

b Next, she wants to write an exponential function whose graph is a horizontal stretch by a factor of

\frac{1}{3}

followed by a reflection in the

x -

axis of the graph of

y = 4^{x} .

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

\frac{1}{2},

the function rule must be multiplied by

\frac{1}{2} .

Function y = 4^{x} Vertical Shrink by a Factor of \frac{1}{2} y = \frac{1}{2} (4)^{x}

Furthermore, a function graph is reflected in the

y -

axis by changing the sign of the input.

Function y = \frac{1}{2} (4)^{x} Reflection in the y - axis y = \frac{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

\frac{1}{3},

the input of the function rule must be multiplied by

\frac{1}{3} .

Function y = 4^{x} Horizontal Stretch by a Factor of \frac{1}{3} y = 4^{\frac{1}{3} x}

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 = 4^{\frac{1}{3} x} Reflection in the x - axis y = - 4^{\frac{1}{3} x}

This can be seen in a coordinate plane.

Explore

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.

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

y = 2^{x} .

Function y = 2^{x} Vertical Translation by k Units 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.

different values

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

Function y = 2^{x} Horizontal Translation by h Units y = 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.

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.

With translations on her mind, Tiffaniqua has drawn the graph of the exponential function $y = 3^{x} .$

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

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

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.

Function y = 3^{x} Translation 2 Units Down y = 3^{x} + (- 2) \Leftrightarrow 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} Translation 1 Unit to the Left y = 3^{x - (- 1)} \Leftrightarrow 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} Translation 2 Units Up and 3 Units to the Right y = 3^{x - 3} + 2

This can be seen on the coordinate plane.

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.

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 $y = (\frac{1}{3})^{x}$ is given.

exponential graph

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

y = \frac{1}{2} (\frac{1}{3})^{x + 1} - 2 .

Answer

Hint

The graph of $y = \frac{1}{2} (\frac{1}{3})^{x + 1} - 2$ is a translation $1$ unit to the left, followed by a vertical shrink by a factor of $\frac{1}{2},$ and a translation $2$ units down of the graph of $y = (\frac{1}{3})^{x} .$

Solution

Consider the given function.

y = \frac{1}{2} (\frac{1}{3})^{x + 1} - 2 ⇕ y = \frac{1}{2} (\frac{1}{3})^{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

\frac{1}{2},

and a translation

2

units down of the graph of

y = (\frac{1}{3})^{x} .

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