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Algebra II : Algebra II

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Example Questions

Example Question #1 : Graphing Logarithmic Functions

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

$f(x) =3 \ln (x+ 4)+ 2$ .

Possible Answers:

y = 2

y = -4

y = 4

y = -2

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

Correct answer:

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

Explanation:

Let $g(x) = \ln x$

In terms of g(x) ,

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

This is the graph of g(x) shifted left 4 units, stretched vertically by a factor of 3, then shifted up 2 units.

The graph of g(x) does not have a horizontal asymptote; therefore, a transformation of this graph, such as that of f(x) , does not have a horizontal asymptote either.

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Example Question #7 : Graphing Logarithmic Functions

Find the equation of the vertical asymptote of the graph of the equation

$f(x) =3 \ln (x+ 4)+ 2$ .

Possible Answers:

x = -2

x = 3

x = 2

x=4

x= -4

Correct answer:

x= -4

Explanation:

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

f(x) =3 g (x+ 4)+ 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 left shift of 4 units (+4) , a vertical stretch ( ), and an upward shift of 2 units ( ). Of the three transformations, only the left shift affects the position of the vertical asymptote - the asymptote of f(x) also shifts left 4 units, to x = -4 .

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Example Question #1 : Negative Exponents

Simplify the following expression

$\left (\frac{f^{-3}}{g^{-2}} \right )^{4}$

Possible Answers:

$\frac{f}{g^{2}}$

$\frac{f^{12}}{g^{8}}$

$\frac{g^{8}}{f^{12}}$

$(fg)^{-4}$

Correct answer:

$\frac{g^{8}}{f^{12}}$

Explanation:

$\left (\frac{f^{-3}}{g^{-2}} \right )^{4}=\left (\frac{g^{2}}{f^{3}} \right )^{4}=\frac{g^{2\times 4}}{f^{3\times4}}=\frac{g^{8}}{f^{12}}$

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Example Question #1 : Exponents

Simplify the following expression

$\left (\frac{34x}{2y} \right )(17xy)^{-1}$

Possible Answers:

$\frac{17}{y^{2}}$

$\frac{1}{y^{2}}$

289x^2

$\frac{x}{y^2}$

Correct answer:

$\frac{1}{y^{2}}$

Explanation:

$\left (\frac{34x}{2y} \right )(17xy)^{-1}=\left (\frac{34x}{2y} \right )\left (\frac{1}{17xy} \right )=\frac{34x}{34xy^2}=\frac{1}{y^2}$

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Example Question #1 : Negative Exponents

Simplify the following expression

$\left (\frac{m}{n} \right )^{-2}\left (\frac{n}{m} \right )^{4}$

Possible Answers:

$\frac{n{^2}}{m^{2}}$

$\frac{n^{8}}{m^{8}}$

$\frac{n^{6}}{m^{6}}$

Correct answer:

$\frac{n^{6}}{m^{6}}$

Explanation:

$\left (\frac{m}{n} \right )^{-2}\left (\frac{n}{m} \right )^{4}=\left (\frac{n}{m} \right )^{2}\left (\frac{n}{m} \right )^{4}=\frac{n^{2+4}}{m^{2+4}}=\frac{n^{6}}{m^{6}}$

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Example Question #2 : Negative Exponents

Simplify the following expression

$\frac{(a^{2})^{-7}b^{0}}{a^{3}b^{-4}}$

Possible Answers:

$a^{14}b^{4}$

$\frac{1}{a^{7}b^{4}}$

$\frac{a^{14}}{b^{4}}$

$\frac{b^{4}}{a^{7}}$

$\frac{b^{4}}{a^{17}}$

Correct answer:

$\frac{b^{4}}{a^{17}}$

Explanation:

$\frac{(a^{2})^{-7}b^{0}}{a^{3}b^{-4}}=\frac{a^{2\times \left(-7\right)}b^0}{a^{3}b^{-4}}=\frac{a^{-14}b^{0}}{a^{3}b^{-4}}=a^{-14-3}b^{0-(-4)}=a^{-17}b^{4}=\frac{b^{4}}{a^{17}}$

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Example Question #3 : Negative Exponents

Solve for :

$(x+5)^{-3} = -1$

Possible Answers:

Correct answer:

Explanation:

Raise both sides of the equation to the inverse power of to cancel the exponent on the left hand side of the equation.

$\rightarrow ((x+5)^{-3})^{-\frac{1}{3}} = (-1)^{-\frac{1}{3}}$

$\rightarrow x+5 = -1$

Subtract from both sides:

$\rightarrow (x+5) - 5 = (-1)-5$

$\rightarrow x = -6$

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Example Question #4 : Negative Exponents

Represent the fraction using only positive exponents:

$\frac{x^{-12}}{x^{-4}y^{-8}}$

Possible Answers:

$y^8x^{16}$

$\frac{y}{x^2}$

$\frac{x^2}{y^2}$

$\frac{y^8}{x^8}$

Correct answer:

$\frac{y^8}{x^8}$

Explanation:

$\frac{x^{-12}}{x^{-4}y^{-8}}$

Negative exponents are the reciprocal of their positive counterpart. For example:

$x^{-2}=\frac{1}{x^2}$

Therefore:

$\frac{x^{-12}}{x^{-4}y^{-8}} = \frac{x^4y^8}{x^{12}}$

This simplifies to:

$\frac{y^8}{x^8}$

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Example Question #7 : Negative Exponents

Solve the equation for n:

$2^{n+4}=\frac{1}{64}$

Possible Answers:

n=-2

n=-10

n=4

n=-12

n=2

Correct answer:

n=-10

Explanation:

Rewrite the right-hand-side so that each side has the same base:

$2^{n+4}=2^{-6}$

Use the Property of Equality for Exponential Functions:

n+4=-6

Solving for :

n=-10

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Example Question #5 : Negative Exponents

What is $5^{-2}$ the same as?

Possible Answers:

$-\frac{1}{25}$

$\frac{1}{25}$

$\frac{1}{5}$

Correct answer:

$\frac{1}{25}$

Explanation:

While a positive exponent says how many times to multiply by a number, a negative exponent says how many times to divide by the number.

To solve for negative exponents, just calculate the reciprocal.

$5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Graphing Logarithmic Functions

Example Question #7 : Graphing Logarithmic Functions

Example Question #1 : Negative Exponents

Example Question #1 : Exponents

Example Question #1 : Negative Exponents

Example Question #2 : Negative Exponents

Example Question #3 : Negative Exponents

Example Question #4 : Negative Exponents

Example Question #7 : Negative Exponents

Example Question #5 : Negative Exponents

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