GMAT Math : Geometry

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 #8 : Graphing 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:

$9\sqrt{3}$

$3\sqrt{3}$

$\sqrt{3}$

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

Example Question #9 : Graphing An Exponential Function

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

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

$\left ( 0,125 \right )$

(0,15)

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

$\left ( 0,243 \right )$

Correct answer:

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

Explanation:

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

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

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

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

Report an Error

Example Question #10 : Graphing An Exponential Function

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

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

Evaluate x+y .

Possible Answers:

The system has no solution.

Correct answer:

Explanation:

Rewrite the system as

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

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

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

2u + v= 59

u- v= -11

Add both sides:

2u + v= 5

$\underline{u- v= 43}$

=48

u = 16 .

Now backsolve:

16- v= -11

- v= - 27

v= 27

Now substitute back:

u = 16

$2 ^{x } = 16$

x = 4

and

v= 27

$3^{y}= 27$

y=3

x+y =4+3 = 7

Report an Error

Example Question #1 : Graphing A Quadratic Function

What are the possible values of if the parabola of the quadratic function $f (x) = Ax^{2} + 4x- 2$ is concave upward and does not intersect the -axis?

Possible Answers:

-2 < A < 4

A > -2

-2 < A < 0

The parabola cannot exist for any value of .

A > 4

Correct answer:

The parabola cannot exist for any value of .

Explanation:

If the graph of $f (x) = Ax^{2} + 4x- 2$ is concave upward, then A > 0 .

If the graph does not intersect the -axis, then $Ax^{2} + 4x- 2 = 0$ has no real solution, and the discriminant $4^{2} - 4 \cdot A \cdot \left ( -2\right )$ is negative:

$4^{2} - 4 \cdot A \cdot \left ( -2\right ) < 0$

16 +8 A < 0

16 +8 A - 16 < 0 - 16

8 A < - 16

$8 A \div 8< - 16 \div 8$

A < -2

For the parabola to have both characteristics, it must be true that A < -2 and A > 0 , but these two events are mutually exclusive. Therefore, the parabola cannot exist.

Report an Error

Example Question #751 : Geometry

Which of the following equations has as its graph a vertical parabola with line of symmetry x = 7 ?

Possible Answers:

$f(x)= 7x^{2}+49x+140$

$f(x)=7x^{2}+98x+140$

$f(x)=7x^{2}-98x+140$

$f(x)=7x^{2} +140$

$f(x)=7x^{2}-49x+140$

Correct answer:

$f(x)=7x^{2}-98x+140$

Explanation:

The graph of $f(x)= Ax^{2}+Bx+ C$ has as its line of symmetry the vertical line of the equation

$x=- \frac{B}{2A}$

Since A=7 in each choice, we want to find such that

$7=- \frac{B}{2A} = - \frac{B}{2(7)} = - \frac{B}{14}$

$- \frac{B}{14} = 7$

$B = 7\cdot (-14)=-98$

so the correct choice is $f(x)=7x^{2}-98x+140$ .

Report an Error

Example Question #1 : Graphing A Quadratic Function

Which of the following equations has as its graph a concave-right horizontal parabola?

Possible Answers:

$y=- 7x^{2}- 4x+13$

$x=7y^{2}- 4y+13$

$x=- 7y^{2}- 4y+13$

$y= 7x^{2}- 4x+13$

None of the other responses gives a correct answer.

Correct answer:

$x=7y^{2}- 4y+13$

Explanation:

A horizontal parabola has as its equation, in standard form,

$x= Ay^{2}+ By+ C$ ,

with A,B,C real, nonzero.

Its orientation depends on the sign of . In the equation of a concave-right parabola, is positive, so the correct choice is $x=7y^{2}- 4y+13$ .

Report an Error

Example Question #1 : Graphing A Quadratic Function

The graphs of the functions f(x) and g(x) have the same line of symmetry.

If we define $f(x) = 2x^{2}+ 4x+7$ , which of the following is a possible definition of g(x) ?

Possible Answers:

$g(x) = 2x^{2}+ 8x+7$

$g(x) = 4x^{2}+8x+7$

$g(x) = x^{2}+ 4x+7$

None of the other responses gives a correct answer.

$g(x) = 2x^{2}-4x-7$

Correct answer:

$g(x) = 4x^{2}+8x+7$

Explanation:

The graph of a function of the form $f(x)= Ax^{2}+Bx+C$ - a quadratic function - is a vertical parabola with line of symmetry $x= - \frac{B}{2A}$ .

The graph of the function $f(x) = 2x^{2}+ 4x+7$ therefore has line of symmetry

$x= - \frac{4}{2 (2)}$ , or x = -1

We examine all four definitions of to find one with this line of symmetry.

$g(x) = 2x^{2}-4x-7$ :

$x= - \frac{-4}{2 (2)}$ , or x = 1

$g(x) = x^{2}+ 4x+7$ :

$x= - \frac{ 4}{2 (1)}$ , or x = -2

$g(x) = 2x^{2}+ 8x+7$

$x= - \frac{ 8}{2 (2)}$ , or x = -2

$g(x) = 4x^{2}+8x+7$

$x= - \frac{ 8}{2 (4)}$ , or x = -1

Since the graph of the function $g(x) = 4x^{2}+8x+7$ has the same line of symmetry as that of the function $f(x) = 2x^{2}+ 4x+7$ , that is the correct choice.

Report an Error

Example Question #1 : How To Graph A Quadratic Function

Give the -coordinate of a point at which the graphs of the equations

$y= x^{2}+ 2x- 7$

and

$y = 2x^{2}- 6x+5$

intersect.

Possible Answers:

Correct answer:

Explanation:

We can set the two quadratic expressions equal to each other and solve for .

$y= x^{2}+ 2x- 7$ and $y = 2x^{2}- 6x+5$ , so

$2x^{2}- 6x+5 = x^{2}+ 2x- 7$

$2x^{2}- 6x+5 -( x^{2}+ 2x- 7)= x^{2}+ 2x- 7 -( x^{2}+ 2x- 7)$

$2x^{2}- 6x+5 - x^{2}- 2x+ 7=0$

$x^{2}- 8x+12=0$

(x-2)(x-6) = 0

The -coordinates of the points of intersection are 2 and 6. To find the -coordinates, substitute in either equation:

$y= x^{2}+ 2x- 7$

$y= 2^{2}+ 2 \cdot 2 - 7$

y= 4+4 - 7

y=1

One point of intersection is (2,1) .

$y= x^{2}+ 2x- 7$

$y= 6^{2}+ 2 \cdot 6 - 7$

y= 36+ 12 - 7

y = 41

The other point of intersection is (6,41) .

1 is not among the choices, but 41 is, so this is the correct response.

Report an Error

Example Question #51 : Coordinate Geometry

Give the set of intercepts of the graph of the function $f(x)= \frac{1}{7} (x-7)^{2}$ .

Possible Answers:

$\left \{ (-7,0), (0,7) \right \}$

$\left \{ (7,0), (0,-7),(0,7) \right \}$

$\left \{ (7,0), (0,7) \right \}$

$\left \{ (7,0), (0,-7) \right \}$

$\left \{ (-7,0), (0,-7),(0,7) \right \}$

Correct answer:

$\left \{ (7,0), (0,7) \right \}$

Explanation:

The -intercepts, if any exist, can be found by setting f(x) = 0 :

$f(x)= \frac{1}{7} (x-7)^{2} = 0$

$\frac{1}{7} (x-7)^{2} \cdot 7 = 0\cdot 7$

$(x-7)^{2} = 0$

x-7 = 0

x= 7

The only -intercept is (7,0) .

The -intercept can be found by substituting 0 for :

$f(0)= \frac{1}{7} (0-7)^{2} = \frac{1}{7} ( -7)^{2}= \frac{1}{7} \cdot 49 = 7$

The -intercept is (0,7) .

The correct set of intercepts is $\left \{ (7,0), (0,7) \right \}$ .

Report an Error

Example Question #7 : Graphing A Quadratic Function

Give the -coordinate of a point of intersection of the graphs of the functions

$f(x) = x^{2}- 4x+ 7$

and

g(x) = 4x .

Possible Answers:

Correct answer:

Explanation:

The system of equations can be rewritten as

$y= x^{2}- 4x+ 7$

y = 4x .

We can set the two expressions in equal to each other and solve:

$x^{2}- 4x+ 7 = 4x$

$x^{2}- 8x+ 7 =0$

(x-1)(x-7)= 0

x= 1,x=7

We can substitute back into the equation y = 4x , and see that either y = 4 or y = 28 . The latter value is the correct choice.

Report an Error

All GMAT Math Resources

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

Popular Subjects

Biology Tutors in Phoenix, Algebra Tutors in San Diego, Math Tutors in San Francisco-Bay Area, Reading Tutors in Los Angeles, LSAT Tutors in Washington DC, Algebra Tutors in Phoenix, SSAT Tutors in San Diego, Reading Tutors in San Diego, Reading Tutors in Denver, Biology Tutors in Boston

Popular Courses & Classes

MCAT Courses & Classes in Miami, LSAT Courses & Classes in Atlanta, LSAT Courses & Classes in Phoenix, LSAT Courses & Classes in San Diego, SAT Courses & Classes in New York City, SAT Courses & Classes in Seattle, SAT Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Washington DC, GRE Courses & Classes in Seattle, GRE Courses & Classes in Denver

Popular Test Prep

SAT Test Prep in Boston, LSAT Test Prep in Phoenix, GRE Test Prep in Atlanta, ISEE Test Prep in San Diego, ISEE Test Prep in Philadelphia, GRE Test Prep in San Francisco-Bay Area, ISEE Test Prep in Chicago, SSAT Test Prep in Atlanta, ISEE Test Prep in Atlanta, GRE Test Prep in Seattle

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 : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #8 : Graphing An Exponential Function

Example Question #9 : Graphing An Exponential Function

Example Question #10 : Graphing An Exponential Function

Example Question #1 : Graphing A Quadratic Function

Example Question #751 : Geometry

Example Question #1 : Graphing A Quadratic Function

Example Question #1 : Graphing A Quadratic Function

Example Question #1 : How To Graph A Quadratic Function

Example Question #51 : Coordinate Geometry

Example Question #7 : Graphing A Quadratic Function

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details