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ACT Math : Graphing

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Example Question #1 : 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:

2π

3π

0.5

Correct answer:

Explanation:

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

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Example Question #1 : How To Graph A Line

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

Possible Answers:

(4, 2)

None of the answers are correct

(2, 4)

(4, 4)

(6, 2)

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 #2 : 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:

$\frac{13}{7}$

$\frac{2}{7}$

$\frac{19}{7}$

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 #24 : Graphing

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 : Graphing Linear Functions

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:

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

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

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

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

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

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 #1 : Graphing Linear Functions

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:

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

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

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

N = -5

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

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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Example Question #1 : How To Graph A Quadratic Function

Best friends John and Elliot are throwing javelins. The height of John’s javelin is described as f(x) = -x²+4x, and the height of Elliot’s javelin is described as f(x) = -2x²+6x, where x is the horizontal distance from the origin of the thrown javelin. Whose javelin goes higher?

Possible Answers:

John’s

Insufficient information provided

Elliot’s

The javelins reach the same height

Correct answer:

Elliot’s

Explanation:

When graphed, each equation is a parabola in the form of a quadratic. Quadratics have the form y = ax² + bx + c, where –b/2a = axis of symmetry. The maximum height is the vertex of each quadratic. Find the axis of symmetry, and plug that x-value into the equation to obtain the vertex.

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Example Question #1 : How To Graph A Quadratic Function

Where does the following equation intercept the x-axis?

y=x^2+7x+12

Possible Answers:

(-4, 0) only

(0, -4) and (0, -3)

(-4, 0) and (-3, 0)

(-4, 0) and (0, -3)

(-7, 0) and (-12, 0)

Correct answer:

(-4, 0) and (-3, 0)

Explanation:

To determine where an equation intercepts a given axis, input 0 for either (where it intercepts the -axis) or (where it intercepts the -axis), then solve. In this case, we want to know where the equation intercepts the -axis; so we will plug in 0 for , giving:

$y = x^{}2 + 7x + 12$

$0 = x^{}2 + 7x + 12$

Now solve for .

Note that in its present form, this is a quadratic equation. In this scenario, we must find two factors of 12, that when added together, equal 7. Quickly, we see that 4 and 3 fit these conditions, giving:

0 = (x + 4)(x + 3)

Solving for , we see that there are two solutions,

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Example Question #3 : How To Graph A Quadratic Function

Where does the following equation intercept the -axis?

y = x^2 - 2x - 3

Possible Answers:

$\left ( -3, 0\right )$ and $\left ( 1, 0\right )$

$\left ( 3, 0\right )$ and $\left ( -1, 0\right )$

$\left ( 3, 0\right )$ and $\left ( 1, 0\right )$

$\left ( 0, 3\right )$ and $\left ( 0, 1\right )$

$\left ( -3, 0\right )$ and $\left ( -1, 0\right )$

Correct answer:

$\left ( 3, 0\right )$ and $\left ( -1, 0\right )$

Explanation:

The x intercept of an equation is the point at which it crosses the x-axis. To find the x intercept, plug in for and solve for .

0 = x^2 - 2x - 3

To solve for , we can factor the equation. We must find two numbers that add to equal and multiply to equal . and fit these conditions, giving:

$0 = \left (x+1\right )\left ( x-3\right )$

We can set each of these equal to to find two solutions for .

0 = x-3 and 0 = x+1

x=3, -1

The x intercepts occur at these values, giving the coordinates:

$\left ( 3, 0\right )$ and $\left ( -1, 0\right )$

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Example Question #4 : How To Graph A Quadratic Function

Where does the following equation intercept the -axis?

y = x^2 + 5x - 9

Possible Answers:

(-1.4, 0) and $\left ( 6.4, 0\right )$

(-1.4, 0) and $\left ( -6.4, 0\right )$

(1.4, 0) and $\left (- 6.4, 0\right )$

(1.4, 0) and $\left (6.4, 0\right )$

(0, 1.4) and $\left (0, -6.4\right )$

Correct answer:

(1.4, 0) and $\left (- 6.4, 0\right )$

Explanation:

The x intercept of an equation is the point at which it crosses the x-axis. To find the x intercept, plug in for and solve for .

0 = x^2 + 5x -9

This equation is not easily factored, so to solve for , we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

With the equation in the form

y = ax^2 + bx + c ,

a = 1 , b = 5 , and c = -9 .

Plugging these values into the quadratic formula, we get:

$x = \frac{-5 \pm \sqrt{5^2 - 4\left (1 \right )\left (-9 \right )}}{2\left (1 \right )} = \frac{-5 \pm \sqrt{61}}{2}$

Find the two solutions for :

$x = \frac{-5 + \sqrt{61}}{2} = 1.4$

$x = \frac{-5 - \sqrt{61}}{2} = -6.4$

The x intercepts occur at these values, giving the coordinates:

(1.4, 0) and $\left (- 6.4, 0\right )$

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ACT Math : Graphing

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 #24 : Graphing

Example Question #1 : Graphing Linear Functions

Example Question #1 : Graphing Linear Functions

Example Question #1 : How To Graph A Quadratic Function

Example Question #1 : How To Graph A Quadratic Function

Example Question #3 : How To Graph A Quadratic Function

Example Question #4 : How To Graph A Quadratic Function

All ACT Math Resources

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