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High School Math : Algebra II

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Example Questions

Example Question #141 : Algebra Ii

The population of a certain bacteria increases exponentially according to the following equation:

$P(t)=2000e^{^{2t}}$

where P represents the total population and t represents time in minutes.

How many minutes does it take for the bacteria's population to reach 48,000?

Possible Answers:

$\frac{\log 24}{2}$

$\frac{2}{\ln 24}$

$\frac{\ln24}{2}$

$\ln 12$

$\frac{2}{\log 24}$

Correct answer:

$\frac{\ln24}{2}$

Explanation:

The question gives us P (48,000) and asks us to find t (time). We can substitute for P and start to solve for t:

$48,000 = 2000e^{2t}$

$24 = e^{2t}$

Now we have to isolate t by taking the natural log of both sides:

$\ln 24 = \ln e^{2t}$

$\ln 24 = (2t)\ln e$

And since $\ln e = 1$ , t can easily be isolated:

$\ln 24 = 2t$

$\frac{\ln24}{2} = t$

Note: $\frac{\ln 24}{2}$ does not equal $\ln 12$ . You have to perform the log operation first before dividing.

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Example Question #11 : Solving Functions

Solve the equation for .

$\small 3^{2x}=81$

Possible Answers:

x=1

x=3

x=9

x=4

x=2

Correct answer:

x=2

Explanation:

Begin by recognizing that both sides of the equation have the same root term, .

$\small 3^{2x}=81$

$\small 3^{2x}=9^2$

$3^{2x}=(3^2)^2$

We can use the power rule to combine exponents.

$3^{2x}=3^4$

Set the exponents equal to each other.

2x=4

$\small x=2$

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Example Question #131 : Mathematical Relationships And Basic Graphs

Solve for :

x^3 - 9x =0

Possible Answers:

x = 0

$x = \frac{1}{3}$

$x = \{-3,0,3\}$

$x = \{-3,3\}$

x = 3

Correct answer:

$x = \{-3,0,3\}$

Explanation:

Pull an out of the left side of the equation.

$\rightarrow x(x^2 - 9) =0$

Use the difference of squares technique to factor the expression in parentheses.

$\rightarrow x(x+3)(x-3) =0$

Any number that causes one of the terms , x+3 , or x-3 to equal is a solution to the equation. These are , , and , respectively.

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Example Question #1 : Graphing Exponential Functions

Find the -intercept(s) of $y=\frac{x^3+2x+8}{x^2-36}$ .

Possible Answers:

$y=\frac{2}{9}$

$y=-\frac{2}{9}$

This function does not cross the -axis.

y=6

y=8

Correct answer:

$y=-\frac{2}{9}$

Explanation:

To find the -intercept, set x=0 in the equation and solve.

$y=\frac{x^3+2x+8}{x^2-36}$

$y=\frac{(0)^3+2(0)+8}{(0)^2-36}$

$y=\frac{(0)+(0)+8}{(0)-36}$

$y=\frac{8}{-36}$

$y=-\frac{2}{9}$

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Example Question #1 : Graphing Exponential Functions

Find the -intercept(s) of $y=\frac{x^2+9x+8}{x^2-44}$ .

Possible Answers:

$y=\frac{2}{11}$

y=-1 and y=-8

$y=-\frac{2}{11}$

$y=2\sqrt{11}$

$y=5\tfrac{1}{2}$

Correct answer:

$y=-\frac{2}{11}$

Explanation:

To find the -intercept(s) of $y=\frac{x^2+9x+8}{x^2-44}$ , set the value equal to zero and solve.

$y=\frac{x^2+9x+8}{x^2-44}$

$y=\frac{(0)^2+9(0)+8}{(0)^2-44}$

$y=\frac{(0)+(0)+8}{(0)-44}$

$y=\frac{8}{-44}$

$y=-\frac{2}{11}$

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Example Question #2 : Graphing Exponential Functions

Find the -intercept(s) of $y=\frac{x^2+9x+8}{x^2-44}$ .

Possible Answers:

x=-1 and x=-8

x=1 and x=8

$x=\frac{1}{8}$

$x=2\sqrt{2}$

$x=2\sqrt{11}$

Correct answer:

x=-1 and x=-8

Explanation:

To find the -intercept(s) of $y=\frac{x^2+9x+8}{x^2-44}$ , we need to set the numerator equal to zero and solve.

First, notice that x^2+9x+8 can be factored into (x+1)(x+8) . Now set that equal to zero: 0=(x+1)(x+8) .

Since we have two sets in parentheses, there are two separate values that can cause our equation to equal zero: one where x+1=0 and one where x+8=0 .

Solve for each value:

x+1=0

x=-1

and

x+8=0

x=-8 .

Therefore there are two -interecpts: x=-1 and x=-8 .

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Example Question #4 : Graphing Exponential Functions

Find the -intercept(s) of $y=\frac{x^3+2x+8}{x^2-36}$ .

Possible Answers:

x=6 or x=-6

x=0

The function does not cross the -axis.

x=-1.67

x=1

Correct answer:

x=-1.67

Explanation:

To find the -intercept(s) of $y=\frac{x^3+2x+8}{x^2-36}$ , we need to set the numerator equal to zero.

That means 0=x^3+2x+8 .

The best way to solve for a funky equation like this is to graph it in your calculator and calculate the roots. The result is $x\approx -1.67$ .

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Example Question #1 : Logarithms

$\log_7118=?$

Possible Answers:

0.06

2.45

16.86

0.56

1.92

Correct answer:

2.45

Explanation:

Most of us don't know what the exponent would be if 7^x=118 and unfortunately there is no $\log_7$ on a graphing calculator -- only $\log$ (which stands for $\log_{10}$ ).

Fortunately we can use the base change rule: $\log_ba=\frac{\log_{10}a}{\log_{10}b}$

Plug in our given values.

$\log_7118=\frac{\log_{10}118}{\log_{10}7}$

$\log_7118=\frac{2.07}{0.845}$

$\log_7118=2.45$

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Example Question #1 : Logarithms

Based on the definition of logarithms, what is ${\log_{10}1000}$ ?

Possible Answers:

100

Correct answer:

Explanation:

For any equation $\log_{10}(y) = x$ , $10^{x } = y$ . Thus, we are trying to determine what power of 10 is 1000. 1000 = 10^3 , so our answer is 3.

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Example Question #2 : Logarithms

What is the value of that satisfies the equation $\log_x128=7$ ?

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

$\log_xa=b$ is equivalent to x^b=a . In this case, you know the value of (the argument of a logarithmic equation) and b (the answer to the logarithmic equation). You must find a solution for the base.

$x^{7}=128$

$\sqrt[7]{x}=\sqrt[7]{128}$

x=2

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High School Math : Algebra II

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #141 : Algebra Ii

Example Question #11 : Solving Functions

Example Question #131 : Mathematical Relationships And Basic Graphs

Example Question #1 : Graphing Exponential Functions

Example Question #1 : Graphing Exponential Functions

Example Question #2 : Graphing Exponential Functions

Example Question #4 : Graphing Exponential Functions

Example Question #1 : Logarithms

Example Question #1 : Logarithms

Example Question #2 : Logarithms

All High School Math Resources

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