GMAT Math : Coordinate Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

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

Example Questions

Example Question #14 : Calculating The Equation Of A Circle

Two circles on the coordinate plane have the origin as their center. The outer circle has twice the area as the inner circle, the equation of which is

$x^{2}+ y^{2} = 5$ .

Give the equation of the outer circle.

Possible Answers:

$x^{2}+ y^{2} = 5 \sqrt{2}$

$x^{2}+ y^{2} = 10$

$x^{2}+ y^{2} = 40$

$x^{2}+ y^{2} = 20$

$x^{2}+ y^{2} = 25$

Correct answer:

$x^{2}+ y^{2} = 10$

Explanation:

The equation of a circle centered at the origin is

$x^{2}+ y^{2} = r^{2}$

where is the radius of the circle. Since the equation of the outer circle is

$x^{2}+ y^{2} = 5$ ,

$r^{2} = 5$ ,

and the area is

$A = \pi r^{2} = \pi \cdot 5 = 5 \pi$ .

The area of the larger circle is twice this, or $2 \cdot 5 \pi = 10 \pi$ ; that is,

$A_{2} = \pi R^{2} = 10 \pi$

$R^{2} = 10$ ,

and the equation of that outer circle is

$x^{2}+ y^{2} = 10$ .

Report an Error

Example Question #15 : Calculating The Equation Of A Circle

Two circles on the coordinate plane have the origin as their center. The outer circle has area five times that of the inner circle; the region between them has area $20 \pi$ . Give the equation of the inner circle.

Possible Answers:

$x^{2}+ y^2 = 4$

$x^{2}+ y^2 = 5$

$x^{2}+ y^2 =16$

$x^{2}+ y^2 = 8$

$x^{2}+ y^2 = 25$

Correct answer:

$x^{2}+ y^2 = 5$

Explanation:

Let be the radius of the inner circle. The area of the inner circle is $A = \pi r^{2}$ ; the outer circle has area five times this, or $A '= 5 \pi r^{2}$ ; the region between them has area equal to the difference of these quantities, or

$A'' = A ' - A= 5 \pi r^{2} - \pi r^{2} = 4 \pi r^{2}$

This is equal to $20 \pi$ , so

$4 \pi r^{2} = 20 \pi$

$\frac{4 \pi r^{2} }{4 \pi}= \frac{20 \pi}{4 \pi }$

$r^{2} =5$

A circle with its center at the origin has as its equation

$x^{2}+ y^2 = r^{2}$ ,

so the inner circle has as its equation

$x^{2}+ y^2 = 5$ .

Report an Error

Example Question #1 : Graphing

Give the -intercept(s) of the graph of the equation

$y = 2^{x } - 20$

Possible Answers:

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

(10,0 )

The graph has no -intercept.

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

(-20,0)

Correct answer:

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

Explanation:

Set y = 0 and solve for :

$y = 2^{x } - 20$

$2^{x } - 20 = 0$

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

$2^{x } = 20$

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

$x = \log_{2} 20$

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

$= 2 + \log_{2} 5$

Report an Error

Example Question #1 : Graphing An Exponential Function

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

(2, 0)

(3,0)

$(\ln 3, 0)$

The graph of has no -intercept.

(0,0)

Correct answer:

The graph of has no -intercept.

Explanation:

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

Report an Error

Example Question #2 : Graphing

Define a function as follows:

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

Give the vertical aysmptote of the graph of .

Possible Answers:

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

x= 2

x= 3

x = 5

The graph of does not have a vertical asymptote.

Correct answer:

The graph of does not have a vertical asymptote.

Explanation:

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

Report an Error

Example Question #3 : Graphing

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

(5,0)

(12,0)

(9,0)

The graph of has no -intercept.

(3,0)

Correct answer:

(5,0)

Explanation:

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

$3 ^{x-3}= 9$

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

x-3 = 2

x = 5

The -intercept is therefore (5,0) .

Report an Error

Example Question #742 : Geometry

Define a function as follows:

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

Give the horizontal aysmptote of the graph of .

Possible Answers:

y = 1

y = 7

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

y = 0

y = 2

Correct answer:

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, $2 ^{x-1} > 0$ and $y= 2 ^{x-1} + 7 > 7$ for all real values of . The graph will never crosst the line of the equatin y = 7 , so this is the horizontal asymptote.

Report an Error

Example Question #143 : Coordinate Geometry

Define functions and as follows:

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

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

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

Possible Answers:

Correct answer:

Explanation:

First, we rewrite both functions with a common base:

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

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

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

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

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

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

f(x) = g(x)

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

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

x+ 2= 2x-2

x+ 2 -x +2= 2x-2 -x +2

x = 4

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

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

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

Report an Error

Example Question #2 : Graphing

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

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

(0,13)

(0,2)

(0,1)

(0,-3)

Correct answer:

(0,13)

Explanation:

The -coordinate ofthe -intercept of the graph of is 0, and its -coordinate is f(0) :

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

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

The -intercept is the point (0,13) .

Report an Error

Example Question #1 : How To Graph An Exponential Function

Define functions and as follows:

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

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

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

Possible Answers:

$\sqrt{3}$

$9\sqrt{3}$

$3\sqrt{3}$

Correct answer:

Explanation:

First, we rewrite both functions with a common base:

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

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

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

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

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

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

f(x) = g(x)

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

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

x+2 = -2x + 2

x+2 + 2x -2 = -2x + 2 + 2x -2

3x=0

x= 0

Now substitute in either function:

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

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

Report an Error

All GMAT Math Resources

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

Popular Subjects

Computer Science Tutors in Denver, LSAT Tutors in Miami, Reading Tutors in Los Angeles, LSAT Tutors in Seattle, English Tutors in Boston, Physics Tutors in Philadelphia, MCAT Tutors in Phoenix, ACT Tutors in Miami, GMAT Tutors in San Francisco-Bay Area, Reading Tutors in Atlanta

Popular Courses & Classes

ACT Courses & Classes in Philadelphia, SAT Courses & Classes in Denver, ACT Courses & Classes in New York City, GRE Courses & Classes in Boston, GMAT Courses & Classes in Atlanta, LSAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in New York City, GMAT Courses & Classes in Houston, MCAT Courses & Classes in Denver, LSAT Courses & Classes in Denver

Popular Test Prep

ISEE Test Prep in Los Angeles, SAT Test Prep in San Diego, LSAT Test Prep in New York City, ACT Test Prep in Phoenix, MCAT Test Prep in Houston, ISEE Test Prep in Phoenix, SAT Test Prep in Phoenix, ACT Test Prep in New York City, ISEE Test Prep in Washington DC, ACT Test Prep in Dallas Fort Worth

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 : Coordinate Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #14 : Calculating The Equation Of A Circle

Example Question #15 : Calculating The Equation Of A Circle

Example Question #1 : Graphing

Example Question #1 : Graphing An Exponential Function

Example Question #2 : Graphing

Example Question #3 : Graphing

Example Question #742 : Geometry

Example Question #143 : Coordinate Geometry

Example Question #2 : Graphing

Example Question #1 : How To Graph An Exponential Function

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details