Coordinate Plane

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ACT Math › Coordinate Plane

Questions 1 - 10

Which of the given functions is depicted below?

Act_math_184_01

Explanation

The graph has x-intercepts at x = 0 and x = 8. This indicates that 0 and 8 are roots of the function.

The function must take the form y = x(x - 8) in order for these roots to be true.

The parabola opens downward, indicating a negative leading coefficient. Expand the equation to get our answer.

y = -x(x - 8)

y = -x2 + 8x

y = 8x - x2

Therefore, the answer must be y = 8x - x2

What is the slope of a line which passes through coordinates $\dpi{100} \small (3,7)$ and $\dpi{100} \small (4,12)$ ?

$\dpi{100} \small 5$

$\dpi{100} \small \frac{1}{5}$

$\dpi{100} \small \frac{1}{2}$

$\dpi{100} \small 2$

$\dpi{100} \small 3$

Explanation

Slope is found by dividing the difference in the $\dpi{100} \small y$ -coordinates by the difference in the $\dpi{100} \small x$ -coordinates.

$\dpi{100} \small \frac{(12-7)}{(4-3)}=\frac{5}{1}=5$

Which of the following lines is perpendicular to the line with the given equation:
?

$y = -\frac{1}{3}x + 5$

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

Explanation

First we must recognize that the equation is given in slope-intercept form, where is the slope of the line.

Two lines are perpendicular if and only if the product of their slopes is . In other words, the slope of a line that is perpendicular to a given line is the negative reciprocal of that slope.

Thus, for a line with a given slope of 3, the line perpendicular to that slope must be the negative reciprocal of 3, or $-\frac{1}{3}$ .

To double check that that does indeed give a product of when multiplied by three simply compute the product:

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

There is a line defined by the equation below:

There is a second line that passes through the point and is parallel to the line given above. What is the equation of this second line?

$y=-\frac{3}{4}x+2.75$

$y=-\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.75$

$y=\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.625$

Explanation

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = _–_3x + 12

y = –(3/4)x + 3

slope = _–_3/4

We know that the second line will also have a slope of _–_3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = _–_3/4(1) + b

2 = _–_3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = –(3/4)x + 2.75

What line is perpendicular to and passes through ?

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

$y = \frac{-3}{5}x + 2$

$y = \frac{1}{3}x - 5$

$y = \frac{2}{5}x + 4$

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

Explanation

Convert the given equation to slope-intercept form.

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

The slope of this line is $-\frac{3}{5}$ . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is $\frac{5}{3}$ .

Plug the new slope and the given point into the slope-intercept form to find the y-intercept.

$2 = \frac{5}{3}(3) + b$

So the equation of the perpendicular line is $y = \frac{5}{3}x - 3$ .

Which of the following lines is perpendicular to the line ?

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

Explanation

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

The slope of this line is . The negative reciprocal will be $+\frac{1}{3}$ , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

$y=\frac{4}{12}x-\frac{6}{12}$

$y=\frac{1}{3}x-\frac{1}{2}$

This equation has a slope of $+\frac{1}{3}$ , and must be our answer.

Which of the following is the equation of a line perpendicular to the line given by:

$y = \frac{1}{3}x + 2$

Explanation

For two lines to be perpendicular their slopes must have a product of .
$-3*\frac{1}{3} = -1$ and so we see the correct answer is given by $y = \frac{1}{3}x + 2$

Which of the given functions is depicted below?

Act_math_184_01

Explanation

The graph has x-intercepts at x = 0 and x = 8. This indicates that 0 and 8 are roots of the function.

The function must take the form y = x(x - 8) in order for these roots to be true.

The parabola opens downward, indicating a negative leading coefficient. Expand the equation to get our answer.

y = -x(x - 8)

y = -x2 + 8x

y = 8x - x2

Therefore, the answer must be y = 8x - x2

Which of the following equations represents a line that is parallel to the line represented by the equation ?

$y=\frac{5}{2}x+1$

$y=-\frac{5}{2}x+1$

$y=\frac{2}{5}x+2$

$y=-\frac{2}{5}x+3$

Explanation

Lines are parallel when their slopes are the same.

First, we need to place the given equation in the slope-intercept form.

Subtract from both sides of the equation.

Simplify.

Divide both sides of the equation by .

$\frac{-4y}{-4}=\frac{-10}{-4}+\frac{26}{-4}$

Simplify.

$y=\frac{10}{4}x-\frac{26}{4}$

Reduce.

$y=\frac{5}{2}x-\frac{13}{2}$

Because the given line has the slope of $\frac{5}{2}$ , the line parallel to it must also have the same slope.

Which of the following is the equation of a line parallel to the line given by the equation:

$y = \frac{1}{4}x +5$

$y = -\frac{1}{4}x + 5$

Explanation

Parallel lines have the same slope and different y-intercepts. If their y-intercepts and slopes are the same they are the same line, and therefore not parallel. Thus the only one that fits the description is:

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