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The equations $\newline$ $2y = -4x + 8$ and $\newline$ $2y = -4x + 16$ $\newline$ are graphed in the $xy$ -plane. Which of the following must be true of the graphs of the two equations? $\newline$ Choose $1$ answer: $\newline$ (A) The graphs of the two equations are parallel lines. $\newline$ (B) The graphs of the two equations are perpendicular lines. $\newline$ (C) The graphs have the same $y$ -intercept. $\newline$ (D) The $y$ -intercepts of the two graphs are reflected in the $x$ -axis.

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Reflections of functions

Full solution

Q. The equations

\newline

2y = -4x + 8

and

\newline

2y = -4x + 16

\newline

are graphed in the

xy

-plane. Which of the following must be true of the graphs of the two equations?

\newline

Choose

1

answer:

\newline

(A) The graphs of the two equations are parallel lines.

\newline

(B) The graphs of the two equations are perpendicular lines.

\newline

(C) The graphs have the same

y

-intercept.

\newline

(D) The

y

-intercepts of the two graphs are reflected in the

x

-axis.

Analyze First Equation: Let's analyze the first equation $2y = -4x + 8$ . $\newline$ To find the slope and y-intercept, we can put it in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. $\newline$ Divide both sides by $2$ to isolate $y$ . $\newline$ $y = -2x + 4$ $\newline$ The slope ( $m$ ) is $-2$ and the y-intercept ( $b$ ) is $y = mx + b$ $0$ .
Analyze Second Equation: Now let's analyze the second equation $2y = -4x + 16$ . Again, we put it in slope-intercept form by dividing both sides by $2$ . $y = -2x + 8$ The slope ( $m$ ) is $-2$ and the y-intercept ( $b$ ) is $8$ .
Comparison of Equations: Comparing the two equations, we see that they both have the same slope, $-2$ , but different y-intercepts, $4$ and $8$ , respectively. $\newline$ Since they have the same slope, the lines are parallel to each other.
Check Answer Choices: Now let's check the answer choices against our findings: $\newline$ (A) The graphs of the two equations are parallel lines. This is true based on our analysis. $\newline$ (B) The graphs of the two equations are perpendicular lines. This is false because perpendicular lines have slopes that are negative reciprocals of each other. $\newline$ (C) The graphs have the same $y$ -intercept. This is false because the $y$ -intercepts are $4$ and $8$ . $\newline$ (D) The $y$ -intercepts of the two graphs are reflected in the $x$ -axis. This is false because reflection in the $x$ -axis would not change the $y$ -intercept's absolute value, only its sign.

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4

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7

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of the following expressions represents the distance between the sandbag and the target?

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\newline

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