Properties of Functions and Graphs

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SAT Math › Properties of Functions and Graphs

Questions 1 - 10
1

Find the slope of the following equation:

Explanation

To find the slope for a given equation, it needs to first be put into the "y=mx+b" format. Then our slope is the number in front of the x, or the "m". For this equation this looks as follows:

First subtract 2x from both sides:

That gives us the following:

Divide all three terms by three to get "y" by itself:

This means our "m" is -2/3

2

Which of the following is NOT a function?

Explanation

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of , there can only be one value of . For the function , there is one value for two possible values. For instance, if , then . But if , as well. This function fails the vertical line test. The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.

3

Find the y-intercept of the following line.

Explanation

To find the y-intercept of any line, we must get the equation into the form

where m is the slope and b is the y-intercept.

To manipulate our equation into this form, we must solve for y. First, we must move the x term to the right side of our equation by subtracting it from both sides.

To isolate y, we now must divide each side by 3.

Now that our equation is in the desired form, our y-intercept is simply

4

What is the vertex of ? Is it a max or min?

Explanation

The polynomial is in standard form of a parabola.

To determine the vertex, first write the formula.

Substitute the coefficients.

Since the is negative is negative, the parabola opens down, and we will have a maximum.

The answer is:

5

Which of the following is NOT a function?

Explanation

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of , there can only be one value of . For the function , there is one value for two possible values. For instance, if , then . But if , as well. This function fails the vertical line test. The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.

6

What is the domain of the following function? Please use interval notation.

Explanation

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.

Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.

If we plug in -100000 for , we get 100000 for .

If we plug in 100000 for , we get 100000 for .

Additionally, if we plug in any value for , we will see that we always get a real, defined value for .

**Extra Note: Due to the absolute value notation, the negative (-) next to the is not important, in that it will always be made positive by the absolute value, making this function the same as . If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding -values negative. However, this knowledge is most important for range, rather than domain.

7

Find the y-intercept of the following line.

Explanation

To find the y-intercept of any line, we must get the equation into the form

where m is the slope and b is the y-intercept.

To manipulate our equation into this form, we must solve for y. First, we must move the x term to the right side of our equation by subtracting it from both sides.

To isolate y, we now must divide each side by 3.

Now that our equation is in the desired form, our y-intercept is simply

8

What is the domain of the following function? Please use interval notation.

Explanation

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.

Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.

If we plug in -100000 for , we get 100000 for .

If we plug in 100000 for , we get 100000 for .

Additionally, if we plug in any value for , we will see that we always get a real, defined value for .

**Extra Note: Due to the absolute value notation, the negative (-) next to the is not important, in that it will always be made positive by the absolute value, making this function the same as . If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding -values negative. However, this knowledge is most important for range, rather than domain.

9

Find the slope of the following equation:

Explanation

To find the slope for a given equation, it needs to first be put into the "y=mx+b" format. Then our slope is the number in front of the x, or the "m". For this equation this looks as follows:

First subtract 2x from both sides:

That gives us the following:

Divide all three terms by three to get "y" by itself:

This means our "m" is -2/3

10

What is the vertex of ? Is it a max or min?

Explanation

The polynomial is in standard form of a parabola.

To determine the vertex, first write the formula.

Substitute the coefficients.

Since the is negative is negative, the parabola opens down, and we will have a maximum.

The answer is:

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