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Advanced Geometry : Coordinate Geometry

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Example Questions

Example Question #121 : Advanced Geometry

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

The graph of has no -intercept.

(3,0)

(5,0)

(9,0)

(12,0)

Correct answer:

(5,0)

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

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

$3 ^{x-3}= 9$

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

x-3 = 2

x = 5

The -intercept is therefore (5,0) .

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Example Question #3 : Graphing

Define a function as follows:

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

Give the horizontal aysmptote of the graph of .

Possible Answers:

y = 1

y = 0

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

y = 7

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.

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Example Question #2 : Graphing

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.

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Example Question #4 : Graphing

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

(0,13)

(0,-3)

(0,2)

(0,1)

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

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) .

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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.

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Example Question #4 : Graphing

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

(0,15)

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

$\left ( 0,125 \right )$

$\left ( 0,243 \right )$

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

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

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Example Question #6 : 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 #71 : Graphing

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest tenth, if applicable.

Possible Answers:

(3.65,0)

The graph has no -interceptx

( -2.92, 0)

( -3.08, 0)

(2.63,0)

Correct answer:

(2.63,0)

Explanation:

The -intercept is (a,0) , where f(a)= 0 :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$5 \cdot 4^{a- 3}- 3 = 0$

$5 \cdot 4^{a- 3}- 3 + 3= 0 + 3$

$5 \cdot 4^{a- 3}= 3$

$5 \cdot 4^{a- 3} \div 5= 3 \div 5$

$4^{a- 3} = 0.6$

$\ln \left (4^{a- 3} \right )=\ln 0.6$

$(a- 3)\ln 4 =\ln 0.6$

$a- 3=\frac{\ln 0.6}{\ln 4 } \approx \frac{-0.5108}{1.3863}\approx -0.3685$

$a- 3+3 \approx -0.3685 + 3$

$a \approx 2.63$

The -intercept is (2.63,0) .

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Example Question #72 : Graphing

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest hundredth, if applicable.

Possible Answers:

(0,3.65)

(0, -3.08)

( 0, 2.63)

The graph has no -intercept

(0, -2.92)

Correct answer:

(0, -2.92)

Explanation:

The -intercept is (0, f(0)) :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$f(0) = 5 \cdot 4^{0- 3}- 3$

$f(0) = 5 \cdot 4^{ - 3}- 3$

$f(0) = 5 \cdot \frac{1}{64}- 3$

$f(0) = \frac{5}{64}- 3$

$f(0) \approx 0.08 - 3 \approx -2.92$

(0, -2.92) is the -intercept.

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Example Question #1 : How To Graph An Exponential Function

Give the vertical asymptote of the graph of the function

$g(x) = 16 \cdot 4^{x}- 3$

Possible Answers:

y = 2

The graph of has no vertical asymptote.

x = 3

y = -2

x = -3

Correct answer:

The graph of has no vertical asymptote.

Explanation:

Since 4 can be raised to the power of any real number, the domain of is the set of all real numbers. Therefore, there is no vertical asymptote of the graph of .

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Advanced Geometry : Coordinate Geometry

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #121 : Advanced Geometry

Example Question #3 : Graphing

Example Question #2 : Graphing

Example Question #4 : Graphing

Example Question #1 : How To Graph An Exponential Function

Example Question #4 : Graphing

Example Question #6 : Graphing

Example Question #71 : Graphing

Example Question #72 : Graphing

Example Question #1 : How To Graph An Exponential Function

All Advanced Geometry Resources

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