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ACT Math : How to graph a line

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Example Questions

Example Question #1 : How To Graph A Line

What is the distance between (7, 13) and (1, 5)?

Possible Answers:

None of the answers are correct

Correct answer:

Explanation:

The distance formula is given by d = square root [(x₂ – x₁)² + (y₂ – y₁)²]. Let point 2 be (7,13) and point 1 be (1,5). Substitute the values and solve.

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Example Question #1 : How To Graph A Line

What is the slope of this line? Screen_shot_2013-07-13_at_5.10.26_pm

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Correct answer:

Explanation:

The slope is found using the formula $m=\frac{y_2-y_1}{x_2-x_1}$ .

We know that the line contains the points (3,0) and (0,6). Using these points in the above equation allows us to calculate the slope.

$\frac{6-0}{0-3}=\frac{6}{-3}=-2$

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Example Question #2 : How To Graph A Line

What is the amplitude of the function if the marks on the y-axis are 1 and -1, respectively?

Screen_shot_2013-07-16_at_10.04.45_am

Possible Answers:

0.5

3π

2π

Correct answer:

Explanation:

The amplitude is half the measure from a trough to a peak.

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Example Question #3 : How To Graph A Line

What is the midpoint between (1, 5) and (7, 3) ?

Possible Answers:

None of the answers are correct

(6, 2)

(4, 2)

(4, 4)

(2, 4)

Correct answer:

(4, 4)

Explanation:

The x-coordinate for the midpoint is given by taking the arithmetic average (mean) of the x-coordinates of the two end points. So the x-coordinate of the midpoint is given by $(1+7)\div2=4$

The same procedure is used for the y-coordinates. So the y-coordinate of the midpoint is given by $(5+3)\div2=4$

Thus the midpoint is given by the ordered pair (4, 4)

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Example Question #4 : How To Graph A Line

If the graph has an equation of y=7x+3 , what is the value of k-n ? Screen_shot_2013-07-16_at_9.41.58_am

Possible Answers:

$\frac{13}{7}$

$\frac{19}{7}$

$\frac{2}{7}$

Correct answer:

$\frac{19}{7}$

Explanation:

is the -intercept and equals . can be solved for by substituting in the equation for , which yields $\frac{2}{7}$

$3-\frac{2}{7}=\frac{21}{7}-\frac{2}{7}=\frac{19}{7}$

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Example Question #5 : How To Graph A Line

The equation 3x + 2y = 6 represents a line. This line does NOT pass through which of the four quadrants?

Possible Answers:

Cannot be determined

III

Correct answer:

III

Explanation:

Plug in for to find a point on the line:

3(0) + 2y = 6

y = 3

Thus, (0,3) is a point on the line.

Plug in for to find a second point on the line:

3x + 2(0) = 6

x = 2

(2,0) is another point on the line.

Now we know that the line passes through the points (2,0) and (0,3) .

A quick sketch of the two points reveals that the line passes through all but the third quadrant.

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Example Question #1 : How To Graph A Function

Line

Refer to the above red line. A line is drawn perpendicular to that line, and with the same -intercept. Give the equation of that line in slope-intercept form.

Possible Answers:

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

$y = \frac{1}{2}x + 8$

$y = - \frac{1}{2}x + 8$

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

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

Correct answer:

$y = - \frac{1}{2}x + 8$

Explanation:

First, we need to find the slope of the above line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-4, y_{1}=x_{2}= 0, y_{2}=8$ :

$m = \frac{8-0}{0-(-4)} = \frac{8}{4} = 2$

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be $m = -\frac{1}{2}$ . Since we want this line to have the same -intercept as the first line, which is the point (0,8) , we can substitute $m = -\frac{1}{2}$ and b = 8 in the slope-intercept form:

y = mx + b

$y = - \frac{1}{2}x + 8$

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Example Question #2 : How To Graph A Function

Axes

Refer to the above diagram. If the red line passes through the point $\left ( N, 4\right )$ , what is the value of ?

Possible Answers:

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

N = -5

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

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

$N = -4\frac{2}{3}$

Correct answer:

$N = -4\frac{2}{3}$

Explanation:

One way to answer this is to first find the equation of the line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-6, y_{1}=x_{2}= 0, y_{2}=18$ :

$m = \frac{18-0}{0-(-6)} = \frac{18}{6} = 3$

The line has slope 3 and -intercept (0,18) , so we can substitute m = 3, b = 18 in the slope-intercept form:

y = mx+b

y = 3x+18

Now substitute 4 for and for and solve for :

4= 3N+18

-14 =3N

$N = -\frac{14}{3}= -4\frac{2}{3}$

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ACT Math : How to graph a line

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Graph A Line

Example Question #1 : How To Graph A Line

Example Question #2 : How To Graph A Line

Example Question #3 : How To Graph A Line

Example Question #4 : How To Graph A Line

Example Question #5 : How To Graph A Line

Example Question #1 : How To Graph A Function

Example Question #2 : How To Graph A Function

All ACT Math Resources

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