GMAT Math : Geometry

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Example Questions

Example Question #751 : Geometry

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:

$3\sqrt{3}$

$9\sqrt{3}$

$\sqrt{3}$

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.

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Example Question #31 : Coordinate Geometry

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

$\left ( 0,243 \right )$

(0,15)

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

$\left ( 0,125 \right )$

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

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

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Example Question #8 : Graphing

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

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

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

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Example Question #141 : Advanced Geometry

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:

A > -2

A > 4

-2 < A < 0

The parabola cannot exist for any value of .

-2 < 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.

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Example Question #12 : Graphing

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

Possible Answers:

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

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

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

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

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

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

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Example Question #142 : Advanced Geometry

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

Possible Answers:

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

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

None of the other responses gives a correct answer.

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

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

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

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Example Question #143 : Advanced Geometry

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:

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

None of the other responses gives a correct answer.

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

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

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

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.

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Example Question #144 : Advanced Geometry

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.

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Example Question #5 : How To Graph A Quadratic Function

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

Possible Answers:

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

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

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

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

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

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

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Example Question #995 : Problem Solving Questions

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.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #751 : Geometry

Example Question #31 : Coordinate Geometry

Example Question #8 : Graphing

Example Question #141 : Advanced Geometry

Example Question #12 : Graphing

Example Question #142 : Advanced Geometry

Example Question #143 : Advanced Geometry

Example Question #144 : Advanced Geometry

Example Question #5 : How To Graph A Quadratic Function

Example Question #995 : Problem Solving Questions

All GMAT Math Resources

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