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SAT II Math I : Solving Functions

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 #1 : Solving Linear Functions

Solve for in terms of .

3y-xy+7=2

Possible Answers:

$y=\frac{5}{x+3}$

$y=\frac{5}{x-3}$

$y=\frac{3}{x-5}$

$y=x+\frac{3}{5}$

$y=\frac{-5}{x-3}$

Correct answer:

$y=\frac{5}{x-3}$

Explanation:

First, subtract 7 to both sides.

3y-xy+7=2

3y-xy=-5

Factor the y on the left hand side.

y(3-x)=-5

Divide both sides by 3 – x.

$y=\frac{-5}{3-x}$

Take an extra step by factoring the minus sign on the denominator.

$y=\frac{-5}{-(x-3)}$

Cancel the minus signs.

$y=\frac{5}{x-3}$

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Example Question #21 : Slope And Line Equations

Find the equation for the line goes through the two points below.

(1,-2)

(-3, 6)

Possible Answers:

y = 4x - 2

y = 2x -4

$y = \frac{1}{2}x$

y = -2x

$y = \frac{1}{2}x + 4$

Correct answer:

y = -2x

Explanation:

Let $P_{1} = (1, \; -2) \; \; and \; \; P_{2} = (-3, \; 6)$ .

First, calculate the slope between the two points.

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{6 - (-2)}{-3 - 1} = \frac{8}{-4} = -2$

Next, use the slope-intercept form to calculate the intercept. We are able to plug in our value for the slope, as well the the values for P_1 .

y = mx + b

-2 = -2(1) + b

b=0

Using slope-intercept form, where we know m=-2 and b=0 , we can see that the equation for this line is y = -2x .

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Example Question #3 : Solving Functions

Find the equation of the line passing through the points $\small (-3,2)$ and $\small (1,8)$ .

Possible Answers:

$\small y=-\frac{3}{2}x+\frac{13}{2}$

$\small y=\frac{2}{3}x-\frac{13}{2}$

$\small y=\frac{2}{3}x+\frac{13}{2}$

$\small y=-\frac{2}{3}x+\frac{13}{2}$

$\small y=\frac{3}{2}x+\frac{13}{2}$

Correct answer:

$\small y=\frac{3}{2}x+\frac{13}{2}$

Explanation:

To calculate a line passing through two points, we first need to calculate the slope, $\small m$ .

$\small m=\frac{\Delta y}{\Delta x} =\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{8-2}{1-(-3)} = \frac{6}{4} = \frac{3}{2}$

Now that we have the slope, we can plug it into our equation for a line in slope intercept form.

$\small y=\frac{3}{2}x+b$

To solve for $\small b$ , we can plug in one of the points we were given. For the sake of this example, let's use $\small (-3,2)$ , but realize either point will give use the same answer.

$\small 2=\frac{3}{2}(-3)+b$

$\small 2=\frac{-9}{2}+b$

$\small 2+\frac{9}{2} = b$

$\small \frac{13}{2} = b$

Now that we have solved for b, we can plug that into our slope intercept form and produce and the answer

$\small y=\frac{3}{2}x+\frac{13}{2}$

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Example Question #2 : Solving Linear Functions

Find the point at which these two lines intersect:

$\small 2x+5y=8$

$\small \small \small -2x+2y=-1$

Possible Answers:

$\left(1,\frac{3}{2}\right)$

$\left(1,-\frac{3}{2}\right)$

$\left(\frac{3}{2},1\right)$

$\left(-\frac{3}{2},1\right)$

Correct answer:

$\left(\frac{3}{2},1\right)$

Explanation:

We are looking for a point, $\small (x,y)$ , where these two lines intersect. While there are many ways to solve for $\small x$ and $\small y$ given two equations, the simplest way I see is to use the elimination method since by adding the two equations together, we can eliminate the $\small x$ variable.

7y=7

Dividing both sides by 7, we isolate y.

$\small y=1$

Now, we can plug y back into either equation and solve for x.

$\small 2x+(5)(1)=8$

$\small 2x+5=8$

Next, we can solve for x.

$\small 2x=3$

$\small x=(3/2)$

Therefore, the point where these two lines intersect is $\left(\frac{3}{2}, 1\right)$ .

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Example Question #4 : Solving Linear Functions

$y=\frac{1}{3}x+9$

Solve for when y=2 .

Possible Answers:

Correct answer:

Explanation:

The first thing is to plug in the given value so you equation is

$2=\frac{1}{3}x+9$ .

Then you must subtract from both sides in order to get by itself.

You now have

$-7=\frac{1}{3}x$

and you must multiply by for each side and you get x=-21 .

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Example Question #4 : Solving Functions

Solve: -7x-16 = 3x+2

Possible Answers:

$-\frac{9}{5}$

$\frac{3}{4}$

$-\frac{9}{2}$

$-\frac{7}{2}$

Correct answer:

$-\frac{9}{5}$

Explanation:

Add on both sides.

-7x-16 +(7x)= 3x+2+(7x)

-16 = 10x+2

Subtract 2 from both sides.

-16 -2= 10x+2-2

-18 = 10x

Divide by 10 on both sides.

$\frac{-18}{10} =\frac{ 10x}{10}$

Reduce the fractions.

The answer is: $-\frac{9}{5}$

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Example Question #1 : Solving Linear Functions

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

Possible Answers:

$\frac{4}{3}$

$\frac{7}{3}$

$\frac{1}{3}$

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=5

y=-5

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

y=3

y=-2

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, x=-2

y = -2, x=2

y = -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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SAT II Math I : Solving Functions

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

All SAT II Math I Resources

Example Questions

Example Question #1 : Solving Linear Functions

Example Question #21 : Slope And Line Equations

Example Question #3 : Solving Functions

Example Question #2 : Solving Linear Functions

Example Question #4 : Solving Linear Functions

Example Question #4 : Solving Functions

Example Question #1 : Solving Linear Functions

Example Question #1 : Solving Exponential Functions

Example Question #1 : Solving Exponential Functions

Example Question #2 : Solving Exponential Functions

All SAT II Math I Resources

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