GMAT Math : Graphing an exponential function

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Graphing

Give the $\displaystyle x$-intercept(s) of the graph of the equation

$y = 2^{x } - 20$ $\displaystyle y = 2^{x } - 20$

Possible Answers:

$\left ( 2 + \log_{2} 5, 0 \right )$ $\displaystyle \left ( 2 + \log_{2} 5, 0 \right )$

(-20,0) $\displaystyle (-20,0)$

The graph has no $\displaystyle x$-intercept.

$\left ( 1 + \log_{2} 5, 0 \right )$ $\displaystyle \left ( 1 + \log_{2} 5, 0 \right )$

(10,0 ) $\displaystyle (10,0 )$

Correct answer:

$\left ( 2 + \log_{2} 5, 0 \right )$ $\displaystyle \left ( 2 + \log_{2} 5, 0 \right )$

Explanation:

Set y = 0 $\displaystyle y = 0$ and solve for $\displaystyle x$:

$y = 2^{x } - 20$ $\displaystyle y = 2^{x } - 20$

$2^{x } - 20 = 0$ $\displaystyle 2^{x } - 20 = 0$

$2^{x } - 20+ 20 = 0 + 20$ $\displaystyle 2^{x } - 20+ 20 = 0 + 20$

$2^{x } = 20$ $\displaystyle 2^{x } = 20$

$\log_{2} \left ( 2 ^{x}\right ) = \log_{2} 20$ $\displaystyle \log_{2} \left ( 2 ^{x}\right ) = \log_{2} 20$

$x = \log_{2} 20$ $\displaystyle x = \log_{2} 20$

$= \log_{2} 4 + \log_{2} 5$ $\displaystyle = \log_{2} 4 + \log_{2} 5$

$= 2 + \log_{2} 5$ $\displaystyle = 2 + \log_{2} 5$

Report an Error

Example Question #1 : How To Graph An Exponential Function

Define a function $\displaystyle f$ as follows:

$f(x) = 3 ^{x-2}$ $\displaystyle f(x) = 3 ^{x-2}$

Give the $\displaystyle x$-intercept of the graph of $\displaystyle f$.

Possible Answers:

(3,0) $\displaystyle (3,0)$

$(\ln 3, 0)$ $\displaystyle (\ln 3, 0)$

(0,0) $\displaystyle (0,0)$

(2, 0) $\displaystyle (2, 0)$

The graph of $\displaystyle f$ has no $\displaystyle x$-intercept.

Correct answer:

The graph of $\displaystyle f$ has no $\displaystyle x$-intercept.

Explanation:

Since the $\displaystyle x$-intercept is the point at which the graph of $\displaystyle f$ intersects the $\displaystyle x$-axis, the $\displaystyle y$-coordinate is 0, and the $\displaystyle x$-coordinate can be found by setting f(x) $\displaystyle f(x)$ equal to 0 and solving for $\displaystyle y$. Therefore, we need to find $\displaystyle x$ such that $3 ^{x-2} = 0$ $\displaystyle 3 ^{x-2} = 0$. However, any power of a positive number must be positive, so $3 ^{x-2} > 0$ $\displaystyle 3 ^{x-2} > 0$ for all real $\displaystyle x$, and $3 ^{x-2} = 0$ $\displaystyle 3 ^{x-2} = 0$ has no real solution. The graph of $\displaystyle f$ therefore has no $\displaystyle x$-intercept.

Report an Error

Example Question #2 : Graphing

Define a function $\displaystyle f$ as follows:

$f(x) = 2 ^{x-3}+ 5$ $\displaystyle f(x) = 2 ^{x-3}+ 5$

Give the vertical aysmptote of the graph of $\displaystyle f$.

Possible Answers:

x= 3 $\displaystyle x= 3$

x= 2 $\displaystyle x= 2$

The graph of $\displaystyle f$ does not have a vertical asymptote.

$x= - \frac{5}{2}$ $\displaystyle x= - \frac{5}{2}$

x = 5 $\displaystyle x = 5$

Correct answer:

The graph of $\displaystyle f$ does not have a vertical asymptote.

Explanation:

Since any number, positive or negative, can appear as an exponent, the domain of the function $f(x) = 2 ^{x-3}+ 5$ $\displaystyle f(x) = 2 ^{x-3}+ 5$ is the set of all real numbers; in other words, f(x) $\displaystyle f(x)$ is defined for all real values of $\displaystyle x$. It is therefore impossible for the graph to have a vertical asymptote.

Report an Error

Example Question #1 : How To Graph An Exponential Function

Define a function $\displaystyle f$ as follows:

$f(x) = 3 ^{x-3}- 9$ $\displaystyle f(x) = 3 ^{x-3}- 9$

Give the $\displaystyle x$-intercept of the graph of $\displaystyle f$.

Possible Answers:

(12,0) $\displaystyle (12,0)$

(5,0) $\displaystyle (5,0)$

(3,0) $\displaystyle (3,0)$

(9,0) $\displaystyle (9,0)$

The graph of $\displaystyle f$ has no $\displaystyle x$-intercept.

Correct answer:

(5,0) $\displaystyle (5,0)$

Explanation:

$3 ^{x-3}- 9 = 0$ $\displaystyle 3 ^{x-3}- 9 = 0$.

$3 ^{x-3}= 9$ $\displaystyle 3 ^{x-3}= 9$

$3 ^{x-3}= 3 ^{2}$ $\displaystyle 3 ^{x-3}= 3 ^{2}$

x-3 = 2 $\displaystyle x-3 = 2$

x = 5 $\displaystyle x = 5$

The $\displaystyle x$-intercept is therefore (5,0) $\displaystyle (5,0)$.

Report an Error

Example Question #5 : Graphing An Exponential Function

Define a function $\displaystyle f$ as follows:

$f(x) = 2 ^{x-1} + 7$ $\displaystyle f(x) = 2 ^{x-1} + 7$

Give the horizontal aysmptote of the graph of $\displaystyle f$.

Possible Answers:

y = 1 $\displaystyle y = 1$

y = 7 $\displaystyle y = 7$

y = 2 $\displaystyle y = 2$

y = 0 $\displaystyle y = 0$

$y = -\frac{7}{2}$ $\displaystyle y = -\frac{7}{2}$

Correct answer:

y = 7 $\displaystyle y = 7$

Explanation:

The horizontal asymptote of an exponential function can be found by noting that a positive number raised to any power must be positive. Therefore, $2 ^{x-1} > 0$ $\displaystyle 2 ^{x-1} > 0$ and $y= 2 ^{x-1} + 7 > 7$ $\displaystyle y= 2 ^{x-1} + 7 > 7$ for all real values of $\displaystyle x$. The graph will never crosst the line of the equatin y = 7 $\displaystyle y = 7$, so this is the horizontal asymptote.

Report an Error

Example Question #6 : Graphing An Exponential Function

Define functions $\displaystyle f$ and $\displaystyle g$ as follows:

$f(x) = 2^{x +2}$ $\displaystyle f(x) = 2^{x +2}$

$g(x) = 4^{x-1}$ $\displaystyle g(x) = 4^{x-1}$

Give the $y \,$ $\displaystyle y \,$-coordinate of the point of intersection of their graphs.

Possible Answers:

$\displaystyle 8$

$\displaystyle 16$

$\displaystyle 4$

$\displaystyle 64$

$\displaystyle 32$

Correct answer:

$\displaystyle 64$

Explanation:

First, we rewrite both functions with a common base:

$f(x) = 2^{x +2}$ $\displaystyle f(x) = 2^{x +2}$ is left as it is.

$g(x) = 4^{x-1}$ $\displaystyle g(x) = 4^{x-1}$ can be rewritten as

$g(x) = \left (2 ^{2} \right )^{x-1}$ $\displaystyle g(x) = \left (2 ^{2} \right )^{x-1}$

$g(x) = 2 ^{2 (x-1)}$ $\displaystyle g(x) = 2 ^{2 (x-1)}$

$g(x) = 2 ^{2x-2}$ $\displaystyle g(x) = 2 ^{2x-2}$

To find the point of intersection of the graphs of the functions, set

$\displaystyle f(x) = g(x)$

$2^{x +2}= 2 ^{2x-2}$ $\displaystyle 2^{x +2}= 2 ^{2x-2}$

The powers are equal and the bases are equal, so we can set the exponents equal to each other and solve:

$\displaystyle x+ 2= 2x-2$

$\displaystyle x+ 2 -x +2= 2x-2 -x +2$

x = 4 $\displaystyle x = 4$

To find the $y\,$ $\displaystyle y\,$-coordinate, substitute 4 for $\displaystyle x$ in either definition:

$f(x) = 2^{x +2}$ $\displaystyle f(x) = 2^{x +2}$

$f(4) = 2^{4+2}= 2^{6} = 64$ $\displaystyle f(4) = 2^{4+2}= 2^{6} = 64$, the correct response.

Report an Error

Example Question #7 : Graphing An Exponential Function

Define a function $\displaystyle f$ as follows:

$f(x) = 4 ^{x+2} - 3$ $\displaystyle f(x) = 4 ^{x+2} - 3$

Give the $\displaystyle y$-intercept of the graph of $\displaystyle f$.

Possible Answers:

(0,13) $\displaystyle (0,13)$

(0,2) $\displaystyle (0,2)$

(0,1) $\displaystyle (0,1)$

(0,-3) $\displaystyle (0,-3)$

$\left ( 0, \frac{3}{4}\right )$ $\displaystyle \left ( 0, \frac{3}{4}\right )$

Correct answer:

(0,13) $\displaystyle (0,13)$

Explanation:

The $\displaystyle x$-coordinate ofthe $\displaystyle y$-intercept of the graph of $\displaystyle f$ is 0, and its $\displaystyle y$-coordinate is f(0) $\displaystyle f(0)$:

$f(x) = 4 ^{x+2} - 3$ $\displaystyle f(x) = 4 ^{x+2} - 3$

$f(0) = 4 ^{0+2} - 3 = 4 ^{2} - 3 = 16 - 3 = 13$ $\displaystyle f(0) = 4 ^{0+2} - 3 = 4 ^{2} - 3 = 16 - 3 = 13$

The $\displaystyle y$-intercept is the point (0,13) $\displaystyle (0,13)$.

Report an Error

Example Question #2 : How To Graph An Exponential Function

Define functions $\displaystyle f$ and $\displaystyle g$ as follows:

$f(x) = 3^{x +2}$ $\displaystyle f(x) = 3^{x +2}$

$g(x) =\left ( \frac{1}{9} \right )^{x-1}$ $\displaystyle g(x) =\left ( \frac{1}{9} \right )^{x-1}$

Give the $y \,$ $\displaystyle y \,$-coordinate of the point of intersection of their graphs.

Possible Answers:

$3\sqrt{3}$ $\displaystyle 3\sqrt{3}$

$9\sqrt{3}$ $\displaystyle 9\sqrt{3}$

$\displaystyle 3$

$\sqrt{3}$ $\displaystyle \sqrt{3}$

$\displaystyle 9$

Correct answer:

$\displaystyle 9$

Explanation:

First, we rewrite both functions with a common base:

$f(x) = 3^{x +2}$ $\displaystyle f(x) = 3^{x +2}$ is left as it is.

$g(x) =\left ( \frac{1}{9} \right )^{x-1}$ $\displaystyle g(x) =\left ( \frac{1}{9} \right )^{x-1}$ can be rewritten as

$g(x) =\left ( 3^{-2} \right )^{x-1}$ $\displaystyle g(x) =\left ( 3^{-2} \right )^{x-1}$

$g(x) = 3^{-2 (x-1)}$ $\displaystyle g(x) = 3^{-2 (x-1)}$

$g(x) = 3^{-2 x+2}$ $\displaystyle g(x) = 3^{-2 x+2}$

To find the point of intersection of the graphs of the functions, set

$\displaystyle f(x) = g(x)$

$3^{x +2} = 3^{-2 x+2}$ $\displaystyle 3^{x +2} = 3^{-2 x+2}$

Since the powers of the same base are equal, we can set the exponents equal:

$\displaystyle x+2 = -2x + 2$

$\displaystyle x+2 + 2x -2 = -2x + 2 + 2x -2$

3x=0 $\displaystyle 3x=0$

x= 0 $\displaystyle x= 0$

Now substitute in either function:

$f(x) = 3^{x +2}$ $\displaystyle f(x) = 3^{x +2}$

$f(0) = 3^{0 +2} = 3^{ 2} = 9$ $\displaystyle f(0) = 3^{0 +2} = 3^{ 2} = 9$, the correct answer.

Report an Error

Example Question #9 : Graphing An Exponential Function

Define a function $\displaystyle f$ as follows:

$f(x) =5^{x-3}$ $\displaystyle f(x) =5^{x-3}$

Give the $\displaystyle y$-intercept of the graph of $\displaystyle f$.

Possible Answers:

$\left ( 0, \frac{1}{243} \right )$ $\displaystyle \left ( 0, \frac{1}{243} \right )$

$\left ( 0,125 \right )$ $\displaystyle \left ( 0,125 \right )$

(0,15) $\displaystyle (0,15)$

$\left ( 0, \frac{1}{125} \right )$ $\displaystyle \left ( 0, \frac{1}{125} \right )$

$\left ( 0,243 \right )$ $\displaystyle \left ( 0,243 \right )$

Correct answer:

$\left ( 0, \frac{1}{125} \right )$ $\displaystyle \left ( 0, \frac{1}{125} \right )$

Explanation:

Since the $\displaystyle y$-intercept is the point at which the graph of $\displaystyle f$ intersects the $\displaystyle y$-axis, the $\displaystyle x$-coordinate is 0, and the $\displaystyle y$-coordinate is f(0) $\displaystyle f(0)$:

$f(x) =5^{x-3}$ $\displaystyle f(x) =5^{x-3}$

$f(0) =5^{0-3}= 5^{-3} = \frac{1}{5^{3}} = \frac{1}{125}$ $\displaystyle f(0) =5^{0-3}= 5^{-3} = \frac{1}{5^{3}} = \frac{1}{125}$,

The $\displaystyle y$-intercept is the point $\left ( 0, \frac{1}{125} \right )$ $\displaystyle \left ( 0, \frac{1}{125} \right )$.

Report an Error

Example Question #10 : Graphing An Exponential Function

$2 ^{x+1} + 3^{y} = 59$ $\displaystyle 2 ^{x+1} + 3^{y} = 59$

$2 ^{x } - 3^{y} = -11$ $\displaystyle 2 ^{x } - 3^{y} = -11$

Evaluate x+y $\displaystyle x+y$.

Possible Answers:

$\displaystyle 43$

$\displaystyle 11$

The system has no solution.

$\displaystyle 7$

$\displaystyle 12$

Correct answer:

$\displaystyle 7$

Explanation:

Rewrite the system as

$2 \cdot 2 ^{x} + 3^{y} = 59$ $\displaystyle 2 \cdot 2 ^{x} + 3^{y} = 59$

$2 ^{x } - 3^{y} = -11$ $\displaystyle 2 ^{x } - 3^{y} = -11$

and substitute $\displaystyle u$ and $\displaystyle v$ for $2 ^{x }$ $\displaystyle 2 ^{x }$ and $3^{y}$ $\displaystyle 3^{y}$, respectively, to form the system

$\displaystyle 2u + v= 59$

$\displaystyle u- v= -11$

Add both sides:

$\displaystyle 2u + v= 5$

$\underline{u- v= 43}$ $\displaystyle \underline{u- v= 43}$

$\displaystyle 3u$ =48 $\displaystyle =48$

u = 16 $\displaystyle u = 16$.

Now backsolve:

$\displaystyle 16- v= -11$

$\displaystyle - v= - 27$

v= 27 $\displaystyle v= 27$

Now substitute back:

u = 16 $\displaystyle u = 16$

$2 ^{x } = 16$ $\displaystyle 2 ^{x } = 16$

x = 4 $\displaystyle x = 4$

and

v= 27 $\displaystyle v= 27$

$3^{y}= 27$ $\displaystyle 3^{y}= 27$

y=3 $\displaystyle y=3$

$\displaystyle x+y =4+3 = 7$

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Calculus Tutors in Dallas Fort Worth, GMAT Tutors in Miami, Reading Tutors in San Diego, SAT Tutors in Denver, ISEE Tutors in Boston, Chemistry Tutors in Atlanta, Reading Tutors in Atlanta, English Tutors in New York City, Calculus Tutors in Phoenix, Statistics Tutors in Los Angeles

Popular Courses & Classes

LSAT Courses & Classes in Philadelphia, ACT Courses & Classes in New York City, SAT Courses & Classes in Houston, ACT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Atlanta, SAT Courses & Classes in Denver, LSAT Courses & Classes in Boston, Spanish Courses & Classes in Atlanta, SSAT Courses & Classes in Seattle, GMAT Courses & Classes in Chicago

Popular Test Prep

MCAT Test Prep in Los Angeles, LSAT Test Prep in Atlanta, ACT Test Prep in New York City, ACT Test Prep in Atlanta, ACT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Chicago, GMAT Test Prep in Boston, SAT Test Prep in Houston, GMAT Test Prep in Miami, SAT Test Prep in San Francisco-Bay Area

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : Graphing an exponential function

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Graphing

Example Question #1 : How To Graph An Exponential Function

Example Question #2 : Graphing

Example Question #1 : How To Graph An Exponential Function

Example Question #5 : Graphing An Exponential Function

Example Question #6 : Graphing An Exponential Function

Example Question #7 : Graphing An Exponential Function

Example Question #2 : How To Graph An Exponential Function

Example Question #9 : Graphing An Exponential Function

Example Question #10 : Graphing An Exponential Function

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details