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College Algebra : Graphs

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Example Questions

Example Question #141 : College Algebra

Give the equation of the horizontal asymptote of the graph of the equation

$f(x) =4 \log (x- 3)- 2$ .

Possible Answers:

y = 4

y = 2

The graph of f(x) does not have a horizontal asymptote.

y = 3

y = 10

Correct answer:

The graph of f(x) does not have a horizontal asymptote.

Explanation:

Let $g(x) = \log x$ . In terms of g(x) ,

f(x) =4 g (x- 3)- 2

g(x) , being a logarithmic function, has a graph without a horizontal asymptote. As f(x) represents the result of transformations of g(x) , it follows that its graph does not have a horizontal asymptote, either.

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Example Question #4 : Transformations

Give the equation of the vertical asymptote of the graph of the equation

$f(x) =4 \log (x- 3)- 2$ .

Possible Answers:

x = 3

x = 4

x = -2

x = 2

x = -3

Correct answer:

x = 3

Explanation:

Let $g(x) = \log x$ . In terms of g(x) ,

f(x) =4 g (x- 3)- 2

The graph of g(x) has as its vertical asymptote the line of the equation x= 0 . The graph of f(x) is the result of three transformations on the graph of g(x) - a right shift of 3 units ( ), a vertical stretch ( ), and a downward shift of 2 units ( ). Of the three transformations, only the right shift affects the position of the vertical asymptote; the asymptote of f(x) also shifts right 3 units, to x = 3 .

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Example Question #5 : Transformations

Define

$f(x) = e^{x}$

and

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

Which two transformations must be performed in the graph of f(x) in order to obtain the graph of g(x) ?

Possible Answers:

None of the other choices gives the correct response.

The graph of f(x) must be translated two units left, then reflected about the -axis.

The graph of f(x) must be translated two units right, then reflected about the -axis.

The graph of f(x) must be translated two units left, then reflected about the -axis.

The graph of f(x) must be translated two units right, then reflected about the -axis.

Correct answer:

The graph of f(x) must be translated two units right, then reflected about the -axis.

Explanation:

g(x) = -f(x-2) , so the graph of g(x) is the result of performing the following transformations:

1) $f_{1}(x) = f(x-2)$ is the result of translating this graph two units right.

2) $g(x)= -f_{1}(x) = -f(x-2)$ is the result of reflecting the new graph about the -axis.

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Example Question #11 : Transformations

The graph of a function f(x) is reflected about the -axis, then translated upward $\frac{8}{3}$ units. Which of the following is represented by the resulting graph?

Possible Answers:

$g(x)= -f\left (-x + \frac{8}{3} \right )$

$g(x)= -f\left (x + \frac{8}{3} \right )$

$g(x)= f(-x) + \frac{8}{3}$

$g(x)= -f\left (x- \frac{8}{3} \right )$

$g(x)= -f(x) + \frac{8}{3}$

Correct answer:

$g(x)= -f(x) + \frac{8}{3}$

Explanation:

Reflecting the graph of a function f(x) about the -axis results in the graph of the function

$f_{1}(x)= -f(x)$ .

Translating this graph upward $\frac{8}{3}$ results in the graph of the function

$g(x)= f_{1}(x) + \frac{8}{3 }= -f(x)+ \frac{8}{3 }$ .

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Example Question #12 : Transformations

Translate the graph of $f(x) = \ln x$ upward three units to yield the graph of a function g(x) . Which of the following is a valid way of stating the definition of g(x) ?

Possible Answers:

$g(x)= \ln (x-3)$

$g(x)= \ln \left ( \frac{x}{e^{3}} \right )$

$g(x)= \ln x^{3}$

$g(x)= \ln (e^{3} x)$

$g(x)= \ln (x+3)$

Correct answer:

$g(x)= \ln (e^{3} x)$

Explanation:

A vertical translation of the graph of a function f(x) by units yields the graph of the function g(x)+k . A translation in an upward direction is a positive translation, so setting $f(x)= \ln x$ and k = 3 , the resulting graph becomes

$g(x)=( \ln x) +3$

$g(x)=3+ \ln x$

Apply properties of logarithms to rewrite this as

$g(x)= \ln e^{3} + \ln x$

$g(x)= \ln (e^{3} x)$ .

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Example Question #11 : Transformations

Reflect the graph of $f(x) = \ln x$ about the -axis to yield the graph of a function g(x) . Which of the following is a valid way of stating the definition of g(x) ?

Possible Answers:

$g(x)= \ln (-x)$

$g(x) =e^{x}$

$g(x)= \ln \frac{1}{x}$

$g(x) = \frac{1}{\ln x}$

None of the other choices gives the correct response.

Correct answer:

$g(x)= \ln (-x)$

Explanation:

The reflection of the graph of a function f(x) about the -axis yields the graph of the function g(x) = f(-x) . Therefore, set $f(x) = \ln x$ and substitute for to yield the function

$g(x) = \ln (-x)$ .

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Example Question #1 : Symmetry

Determine the symmetry of the following equation.

y=x^2-6x+10

Possible Answers:

Symmetry along the x-axis.

Symmetry along the origin.

Does not have symmetry.

Symmetry along the y-axis.

Symmetry along all axes.

Correct answer:

Does not have symmetry.

Explanation:

To check for symmetry, we are going to do three tests, which involve substitution. First one will be to check symmetry along the x-axis, replace -y=y .

-y=x^2-6x+10

This isn't equivilant to the first equation, so it's not symmetric along the x-axis.

Next is to substitute -x=x .

y=(-x)^2-6(-x)+10

y=x^2+6x+10

This is not the same, so it is not symmetric along the y-axis.

For the last test we will substitute -y=y , and -x=x

-y=(-x)^2-6(-x)+10

-y=x^2+6x+10

y=-x^2-6x-10

This isn't the same as the orginal equation, so it is not symmetric along the origin.

The answer is it is not symmetric along any axis.

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Example Question #2 : Symmetry

Untitled

Which of the following is true of the relation graphed above?

Possible Answers:

It is a function, but it is neither even nor odd.

It is an even function

It is not a function

It is an odd function

Correct answer:

It is an odd function

Explanation:

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Untitled

Also, it can be seen to be symmetrical about the origin. Consequently, for each in the domain, f(-c) = -f(c) - the function is odd.

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Example Question #3 : Symmetry

Relation

Which of the following is true of the relation graphed above?

Possible Answers:

It is a function, but it is neither even nor odd.

It is not a function

It is an even function

It is an odd function

Correct answer:

It is an odd function

Explanation:

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Relation

Also, it is seen to be symmetric about the origin. Consequently, for each in the domain, f(-c) = -f(c) - the function is odd.

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Example Question #4 : Symmetry

f(x) is an even function; f(9) = 10 .

True or false: It follows that f(10) = 9 .

Possible Answers:

False

True

Correct answer:

False

Explanation:

A function f(x) is even if and only if, for all in its domain, f(-x) = f(x) . It follows that if f(9) = 10 , then

f(-9)= f(9) = 10 .

No restriction is placed on any other value as a result of this information, so the answer is false.

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College Algebra : Graphs

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #141 : College Algebra

Example Question #4 : Transformations

Example Question #5 : Transformations

Example Question #11 : Transformations

Example Question #12 : Transformations

Example Question #11 : Transformations

Example Question #1 : Symmetry

Example Question #2 : Symmetry

Example Question #3 : Symmetry

Example Question #4 : Symmetry

All College Algebra Resources

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