<< Chapter < Page Chapter >> Page >

Section exercises

Verbal

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

An asymptote is a line that the graph of a function approaches, as x either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

Got questions? Get instant answers now!

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

Got questions? Get instant answers now!

Algebraic

The graph of f ( x ) = 3 x is reflected about the y -axis and stretched vertically by a factor of 4. What is the equation of the new function, g ( x ) ? State its y -intercept, domain, and range.

g ( x ) = 4 ( 3 ) x ; y -intercept: ( 0 , 4 ) ; Domain: all real numbers; Range: all real numbers greater than 0.

Got questions? Get instant answers now!

The graph of f ( x ) = ( 1 2 ) x is reflected about the y -axis and compressed vertically by a factor of 1 5 . What is the equation of the new function, g ( x ) ? State its y -intercept, domain, and range.

Got questions? Get instant answers now!

The graph of f ( x ) = 10 x is reflected about the x -axis and shifted upward 7 units. What is the equation of the new function, g ( x ) ? State its y -intercept, domain, and range.

g ( x ) = 10 x + 7 ; y -intercept: ( 0 , 6 ) ; Domain: all real numbers; Range: all real numbers less than 7.

Got questions? Get instant answers now!

The graph of f ( x ) = ( 1.68 ) x is shifted right 3 units, stretched vertically by a factor of 2 , reflected about the x -axis, and then shifted downward 3 units. What is the equation of the new function, g ( x ) ? State its y -intercept (to the nearest thousandth), domain, and range.

Got questions? Get instant answers now!

The graph of f ( x ) = 2 ( 1 4 ) x 20 is shifted left 2 units, stretched vertically by a factor of 4 , reflected about the x -axis, and then shifted downward 4 units. What is the equation of the new function, g ( x ) ? State its y -intercept, domain, and range.

g ( x ) = 2 ( 1 4 ) x ; y -intercept: ( 0 ,  2 ) ; Domain: all real numbers; Range: all real numbers greater than 0.

Got questions? Get instant answers now!

Graphical

For the following exercises, graph the function and its reflection about the y -axis on the same axes, and give the y -intercept.

g ( x ) = 2 ( 0.25 ) x

Graph of two functions, g(-x)=-2(0.25)^(-x) in blue and g(x)=-2(0.25)^x in orange.

y -intercept: ( 0 , 2 )

Got questions? Get instant answers now!

h ( x ) = 6 ( 1.75 ) x

Got questions? Get instant answers now!

For the following exercises, graph each set of functions on the same axes.

f ( x ) = 3 ( 1 4 ) x , g ( x ) = 3 ( 2 ) x , and h ( x ) = 3 ( 4 ) x

Graph of three functions, g(x)=3(2)^(x) in blue, h(x)=3(4)^(x) in green, and f(x)=3(1/4)^(x) in orange.
Got questions? Get instant answers now!

f ( x ) = 1 4 ( 3 ) x , g ( x ) = 2 ( 3 ) x , and h ( x ) = 4 ( 3 ) x

Got questions? Get instant answers now!

For the following exercises, match each function with one of the graphs in [link] .

Graph of six exponential functions.

f ( x ) = 2 ( 0.69 ) x

B

Got questions? Get instant answers now!

f ( x ) = 2 ( 0.81 ) x

A

Got questions? Get instant answers now!

f ( x ) = 2 ( 1.59 ) x

E

Got questions? Get instant answers now!

For the following exercises, use the graphs shown in [link] . All have the form f ( x ) = a b x .

Graph of six exponential functions.

Which graph has the largest value for b ?

D

Got questions? Get instant answers now!

Which graph has the smallest value for b ?

Got questions? Get instant answers now!

Which graph has the largest value for a ?

C

Got questions? Get instant answers now!

Which graph has the smallest value for a ?

Got questions? Get instant answers now!

For the following exercises, graph the function and its reflection about the x -axis on the same axes.

f ( x ) = 3 ( 0.75 ) x 1

Got questions? Get instant answers now!

f ( x ) = 4 ( 2 ) x + 2

Graph of two functions, -f(x)=(4)(2)^(x)-2 in blue and f(x)=(-4)(2)^x+1 in orange.
Got questions? Get instant answers now!

For the following exercises, graph the transformation of f ( x ) = 2 x . Give the horizontal asymptote, the domain, and the range.

h ( x ) = 2 x + 3

Graph of h(x)=2^(x)+3.

Horizontal asymptote: h ( x ) = 3 ; Domain: all real numbers; Range: all real numbers strictly greater than 3.

Got questions? Get instant answers now!

For the following exercises, describe the end behavior of the graphs of the functions.

f ( x ) = 5 ( 4 ) x 1

As x , f ( x ) ;
As x , f ( x ) 1

Got questions? Get instant answers now!

f ( x ) = 3 ( 1 2 ) x 2

Got questions? Get instant answers now!

f ( x ) = 3 ( 4 ) x + 2

As x , f ( x ) 2 ;
As x , f ( x )

Got questions? Get instant answers now!

For the following exercises, start with the graph of f ( x ) = 4 x . Then write a function that results from the given transformation.

Shift f ( x ) 4 units upward

Got questions? Get instant answers now!

Shift f ( x ) 3 units downward

f ( x ) = 4 x 3

Got questions? Get instant answers now!

Shift f ( x ) 2 units left

Got questions? Get instant answers now!

Shift f ( x ) 5 units right

f ( x ) = 4 x 5

Got questions? Get instant answers now!

Reflect f ( x ) about the x -axis

Got questions? Get instant answers now!

Reflect f ( x ) about the y -axis

f ( x ) = 4 x

Got questions? Get instant answers now!

For the following exercises, each graph is a transformation of y = 2 x . Write an equation describing the transformation.

For the following exercises, find an exponential equation for the graph.

Numeric

For the following exercises, evaluate the exponential functions for the indicated value of x .

g ( x ) = 1 3 ( 7 ) x 2 for g ( 6 ) .

g ( 6 ) = 800 + 1 3 800.3333

Got questions? Get instant answers now!

f ( x ) = 4 ( 2 ) x 1 2 for f ( 5 ) .

Got questions? Get instant answers now!

h ( x ) = 1 2 ( 1 2 ) x + 6 for h ( 7 ) .

h ( 7 ) = 58

Got questions? Get instant answers now!

Technology

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f ( x ) = a b x + d .

50 = ( 1 2 ) x

Got questions? Get instant answers now!

116 = 1 4 ( 1 8 ) x

x 2.953

Got questions? Get instant answers now!

5 = 3 ( 1 2 ) x 1 2

x 0.222

Got questions? Get instant answers now!

30 = 4 ( 2 ) x + 2 + 2

Got questions? Get instant answers now!

Extensions

Explore and discuss the graphs of F ( x ) = ( b ) x and G ( x ) = ( 1 b ) x . Then make a conjecture about the relationship between the graphs of the functions b x and ( 1 b ) x for any real number b > 0.

The graph of G ( x ) = ( 1 b ) x is the refelction about the y -axis of the graph of F ( x ) = b x ; For any real number b > 0 and function f ( x ) = b x , the graph of ( 1 b ) x is the the reflection about the y -axis, F ( x ) .

Got questions? Get instant answers now!

Prove the conjecture made in the previous exercise.

Got questions? Get instant answers now!

Explore and discuss the graphs of f ( x ) = 4 x , g ( x ) = 4 x 2 , and h ( x ) = ( 1 16 ) 4 x . Then make a conjecture about the relationship between the graphs of the functions b x and ( 1 b n ) b x for any real number n and real number b > 0.

The graphs of g ( x ) and h ( x ) are the same and are a horizontal shift to the right of the graph of f ( x ) ; For any real number n , real number b > 0 , and function f ( x ) = b x , the graph of ( 1 b n ) b x is the horizontal shift f ( x n ) .

Got questions? Get instant answers now!

Prove the conjecture made in the previous exercise.

Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask