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SAT II Math I : Functions and Graphs

Study concepts, example questions & explanations for SAT II Math I

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All SAT II Math I Resources

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Example Questions

Example Question #7 : Solving Linear Functions

Solve: 3x+5 = -2(2-3x)

Possible Answers:

$\frac{1}{3}$

$\frac{7}{3}$

$\frac{4}{3}$

Correct answer:

Explanation:

Distribute the right side.

-2(2-3x) = -4 +6x

Rewrite the equation.

3x+5 = -4+6x

Subtract on both sides.

3x+5 -3x= -4+6x-3x

5=-4+3x

Add 4 on both sides.

5+4=-4+3x+4

9=3x

Divide by three on both sides.

$\frac{9}{3}=\frac{3x}{3}$

The answer is:

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Example Question #1 : Solving Exponential Functions

Solve the following function:

$11=x^{2}-5$

Possible Answers:

and

Correct answer:

and

Explanation:

You must get by itself so you must add to both side which results in

$16=x^{2}$ .

You must get the square root of both side to undue the exponent.

This leaves you with x=4 .

But since you square the in the equation, the original value you plug can also be its negative value since squaring it will make it positive anyway.

This means your answer can be x=4 or x=-4 .

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Example Question #1 : Solving Exponential Functions

What is the horizontal asymptote of the graph of the equation $y = -2e^{x-3} -5$ ?

Possible Answers:

$y=-\frac{5}{2}$

y=-5

y=-2

y=3

y=5

Correct answer:

y=-5

Explanation:

The asymptote of this equation can be found by observing that $e^{x-3} > 0$ regardless of . We are thus solving for the value of as $e^{x-3}$ approaches zero.

$-2e^{x-3} < -2\cdot 0$

$-2e^{x-3} < 0$

$-2e^{x-3} -5 < 0-5$

$-2e^{x-3} -5 < -5$

y<-5

So the value that cannot exceed is , and the line y = -5 is the asymptote.

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Example Question #2 : Solving Exponential Functions

What is/are the asymptote(s) of the graph of the function

$f(x) = 5e^{x-2}-2$ ?

Possible Answers:

x=2

y = -2

y = -2, x=2

y = -2, x=-2

x=-2

Correct answer:

y = -2

Explanation:

An exponential equation of the form $f(x) = ae^{x-h} +k$ has only one asymptote - a horizontal one at y = k . In the given function, k = -2 , so its one and only asymptote is y = -2 .

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Example Question #2 : Solving Exponential Functions

Find the vertical asymptote of the equation.

$y=\frac{x^2-9}{x^2-16}$

Possible Answers:

x=4

x=4, -4

There are no vertical asymptotes.

x=1

x=0

Correct answer:

x=4, -4

Explanation:

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

0=x^2-16

16=x^2

$\sqrt{16}=\sqrt{x^2}$

4, -4=x

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Example Question #1 : Asymptotes

Consider the exponential function y=2^x+3 . Determine if there are any asymptotes and where they lie on the graph.

Possible Answers:

There are no asymptotes. goes to positive infinity in both the and directions.

There is one vertical asymptote at y=2 .

There is one horizontal asymptote at y=3 .

There is one vertical asymptote at y=3 .

Correct answer:

There is one horizontal asymptote at y=3 .

Explanation:

For positive values, increases exponentially in the direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative values, as decreases, the term 2^x becomes closer and closer to zero so approaches as we move along the negative axis. As the graph below shows, this is forms a horizontal asymptote.

Exp_asymp

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Example Question #51 : Functions And Graphs

Solve the equation for .

$\small 9^x=3^6$

Possible Answers:

$\small x=2$

$\small x=0$

$\small x=1$

$\small x=3$

Correct answer:

$\small x=3$

Explanation:

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

$\small 9^x=3^6$

(3^2)^x=3^6

Using the power rule, we can set the exponents equal to each other.

$3^{(2*x)}=3^6$

$\small 2x=6$

$\small x=3$

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Example Question #3 : Solving Exponential Functions

Solve the equation for .

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

Possible Answers:

x=3

x=1

x=4

x=9

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 #1 : Solving Exponential Equations

In 2009, the population of fish in a pond was 1,034. In 2013, it was 1,711.

Write an exponential growth function of the form $y=ab^{x}}$ that could be used to model , the population of fish, in terms of , the number of years since 2009.

Possible Answers:

y=1034(1.1342)^x

y=1.35(1.0011)^x

y=1.2(1711)^x

y=1711(1.3932)^x

y=1.1(1034)^x

Correct answer:

y=1034(1.1342)^x

Explanation:

Solve for the values of a and b:

In 2009, y=1034 and x=0 (zero years since 2009). Plug this into the exponential equation form:

$1034=ab^{(0)}$ . Solve for to get .

In 2013, y=1711 and x=4 . Therefore,

1711=ab^4 or 1711=(1034)b^4 . Solve for to get

$b=\sqrt[4]{\frac{1711}{1034}}\approx1.1342$ .

Then the exponential growth function is

$y=1034(1.1342)^{x}$ .

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Example Question #1 : Solving Exponential Functions

$4^{5x}=8^{4x-1}$

Solve for .

Possible Answers:

1.5

Correct answer:

1.5

Explanation:

8 and 4 are both powers of 2.

$4^{5x}=8^{4x-1}$

$2^{10x}=2^{12x-3}$

10x=12x-3 -2x=-3 x=1.5

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SAT II Math I : Functions and Graphs

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #7 : Solving Linear Functions

Example Question #1 : Solving Exponential Functions

Example Question #1 : Solving Exponential Functions

Example Question #2 : Solving Exponential Functions

Example Question #2 : Solving Exponential Functions

Example Question #1 : Asymptotes

Example Question #51 : Functions And Graphs

Example Question #3 : Solving Exponential Functions

Example Question #1 : Solving Exponential Equations

Example Question #1 : Solving Exponential Functions

All SAT II Math I Resources

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