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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #3881 : Algebra Ii

Solve: $2^6 = (\frac{1}{8})^{3x+3}$

Possible Answers:

$-\frac{17}{3}$

$\frac{55}{9}$

$-\frac{5}{8}$

$-\frac{5}{3}$

$\frac{1}{3}$

Correct answer:

$-\frac{5}{3}$

Explanation:

Change the base of the fraction to base 2.

$\frac{1}{8} =2^{-3}$

$2^6 = (2^{-3})^{3x+3}$

Now that both bases are similar, we can set the powers equal to each other.

6 = -3(3x+3)

Use distribution to simplify the right side.

6=-9x-9

Add 9 on both sides.

6+9=-9x-9+9

15=-9x

Divide by negative nine on both sides.

$\frac{15}{-9}=\frac{-9x}{-9}$

$-\frac{15}{9}$

Reduce this fraction.

The answer is: $-\frac{5}{3}$

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Example Question #104 : Solving Exponential Equations

Evaluate: $2^{8x+6} = 16^{30}$

Possible Answers:

$\frac{71}{4}$

$\frac{57}{4}$

$\frac{13}{4}$

$\frac{11}{5}$

Correct answer:

$\frac{57}{4}$

Explanation:

Change the base of the second number to base two.

16= 2^4

We can then replace the term and use the power rule of exponents to simplify the equation.

$2^{8x+6} = 2^{4(30)}$

Set the powers equal now that we have same bases.

8x+6 = 120

Subtract 6 from both sides.

8x+6 -6= 120-6

8x= 114

Divide by eight on both sides.

$\frac{8x}{8}= \frac{114}{8}$

Reduce the fractions.

The answer is: $\frac{57}{4}$

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Example Question #112 : Solving And Graphing Exponential Equations

Evaluate: $e= (\frac{1}{e^2})^{3x-3}$

Possible Answers:

$\textup{No solution.}$

$\frac{5}{6}$

$\frac{2}{3}$

Correct answer:

$\frac{5}{6}$

Explanation:

Convert the bases to to some power on both sides.

e= e^1

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

Rewrite the terms of the equation.

$e^1= (e^{-2})^{3x-3}$

With similar bases, we can set the powers equal to each other.

1=(-2)(3x-3)

Simplify the right side by distribution.

1=-6x+6

Subtract six on both sides, and divide by negative six.

1-6=-6x+6-6

5=-6x

$\frac{5}{-6}=\frac{-6x}{-6}$

The answer is: $\frac{5}{6}$

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Example Question #111 : Solving And Graphing Exponential Equations

Solve: $(2x-3)^{\frac{1}{4}} =3$

Possible Answers:

$\textup{No solution.}$

Correct answer:

Explanation:

We can rewrite the fractional exponent as a radical.

$\sqrt[4]{2x-3}=3$

Raise both sides by the power of four to eliminate the radical.

$(\sqrt[4]{2x-3} )^4=3^4$

2x-3 = 81

Add three on both sides.

2x-3+3 = 81+3

2x=84

Divide by two on both sides.

$\frac{2x}{2}=\frac{84}{2}$

Reduce both fractions.

The answer is:

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Example Question #113 : Solving And Graphing Exponential Equations

Solve: $\frac{1}{3} = 81^{-3x+5}$

Possible Answers:

$\frac{19}{12}$

$\frac{7}{4}$

$-\frac{19}{12}$

$\frac{7}{6}$

$\frac{7}{3}$

Correct answer:

$\frac{7}{4}$

Explanation:

To be able to solve this equation, we will need to change the bases on both sides to a common base.

Choose three as the common base.

$\frac{1}{3} =3^{-1}$

81=3^4

Replace the terms into the equation.

$3^{-1} =3^{4(-3x+5)}$

With the common bases, we can set the powers equal to each other.

-1 = 4(-3x+5)

Distribute the four on the right side.

-1 = -12x+20

Subtract 20 on both sides.

-1-20 = -12x+20-20

-21 = -12x

Divide by negative 12 on both sides.

$\frac{-21 }{-12}= \frac{-12x}{12}$

Reduce both fractions.

$\frac{-21 }{-12} = \frac{7\times 3}{4\times 3}$

The answer is: $\frac{7}{4}$

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Example Question #108 : Solving Exponential Equations

Solve: $4^{x+10} = (\sqrt2)^{-2x+3}$

Possible Answers:

$-\frac{37}{6}$

$-\frac{17}{4}$

$\textup{No solution.}$

$-\frac{43}{6}$

$-\frac{23}{4}$

Correct answer:

$-\frac{37}{6}$

Explanation:

Convert the bases to a common base so that the exponential powers can be set equal to each other. We can choose base two and rewrite both sides using exponential powers.

The radical is to the power of one-half.

$(2^2)^{x+10} = (2^{\frac{1}{2}})^{(-2x+3)}$

Set the powers equal to each other.

$2(x+10) =\frac{1}{2}(-2x+3)$

Distribute the outer terms into the binomials.

$2x+20 = -x+\frac{3}{2}$

Multiply by two on both sides to eliminate the fraction.

$2(2x+20) =2( -x+\frac{3}{2})$

4x+40 = -2x+3

Add on both sides.

4x+40+2x = -2x+3+2x

6x+40 = 3

Subtract 40 on both sides.

6x+40-40 = 3-40

6x=-37

Divide by 6 on both sides.

$\frac{6x}{6}=\frac{-37}{6}$

The answer is: $-\frac{37}{6}$

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Example Question #1 : Graphing Exponential Functions

Give the -intercept of the graph of the equation $f(x) = 2 ^{x- 7} + 4$ .

Possible Answers:

(5,0)

(3.9921875, 0)

(4.0078125, 0)

(9,0)

The graph has no -intercept.

Correct answer:

The graph has no -intercept.

Explanation:

Set f(x) = 0 and solve for

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

$2 ^{x- 7} + 4 = 0$

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

We need not work further. It is impossible to raise a positive number 2 to any real power to obtain a negative number. Therefore, the equation has no solution, and the graph of has no -intercept.

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Example Question #21 : Solving Exponential Functions

What is/are the asymptote(s) of the graph of the function $g(x) = 0.4 ^{x - 2} + 0.7$ ?

Possible Answers:

y = 0.4

y = 1.1

y = 0.7

x = 2

x = 2 and y = 0.7

Correct answer:

y = 0.7

Explanation:

An exponential function of the form

$g(x) = a^{x- h}+k$

has as its one and only asymptote the horizontal line y = k .

Since we define as

$g(x) = 0.4 ^{x - 2} + 0.7$ ,

then k= 0.7 ,

and the only asymptote is the line of the equation y = 0.7 .

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Example Question #21 : Solving Exponential Functions

Determine whether each function represents exponential decay or growth.

$\newline a) \newline y=(\frac{1}{2})^{x}$

$\newline b) \newline y=0.4(1.1)^x$

Possible Answers:

a) growth

b) decay

a) decay

b) growth

a) growth

b) growth

a) decay

b) decay

Correct answer:

a) decay

b) growth

Explanation:

This is exponential decay since the base, (1/2) , is between and .

This is exponential growth since the base, 1.1 , is greater than .

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Example Question #4 : Graphing Exponential Functions

Match each function with its graph.

1. y=2^x

2. y=3(2^x)

3. y=(1/2)^x

a. 3time2tothex

b. 1over2tothex

c. 2_tothe_x

Possible Answers:

Correct answer:

Explanation:

For y=2^x , our base is greater than so we have exponential growth, meaning the function is increasing. Also, when x=0 , we know that y=1 since 2^0=1 . The only graph that fits these conditions is .

For y=3(2^x) , we have exponential growth again but when x=0 , y=3 . This is shown on graph .

For y=(1/2)^x , we have exponential decay so the graph must be decreasing. Also, when x=0 , y=1 . This is shown on graph .

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3881 : Algebra Ii

Example Question #104 : Solving Exponential Equations

Example Question #112 : Solving And Graphing Exponential Equations

Example Question #111 : Solving And Graphing Exponential Equations

Example Question #113 : Solving And Graphing Exponential Equations

Example Question #108 : Solving Exponential Equations

Example Question #1 : Graphing Exponential Functions

Example Question #21 : Solving Exponential Functions

Example Question #21 : Solving Exponential Functions

Example Question #4 : Graphing Exponential Functions

All Algebra II Resources

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