<< 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!

Questions & Answers

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask