Question

Calculate the equation of a line that is perpendicular to the line shown on the graph and that also shares the same y-intercept.
First point: (-8, 0)
Second point: (0, 4)
A. y = -1/2x + 4
B. y = 2x - 8
C. y = 1/2x - 8
D. y = -2x + 4

Answer 1

To find the equation of the line that is perpendicular to the given line and shares the same y-intercept, we first need to determine the slope of the line that passes through the points (-8, 0) and (0, 4).

1. Calculate the slope (m) of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \((x_1, y_1) = (-8, 0)\) and \((x_2, y_2) = (0, 4)\).

\[
m = \frac{4 - 0}{0 - (-8)} = \frac{4}{8} = \frac{1}{2}
\]

2. The slope of the line that is perpendicular to this line will be the negative reciprocal of \(\frac{1}{2}\):
\[
m_{\text{perpendicular}} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]

3. Since we want the new line to share the same y-intercept as the original line, we find the y-intercept from the original line. We can determine the original line's equation in slope-intercept form (y = mx + b). We already have \(m = \frac{1}{2}\) and the y-intercept (b) from the point (0, 4), which is \(b = 4\).

4. Now we can use the slope-intercept form to write the equation of the perpendicular line with the slope \(-2\) and y-intercept \(4\):
\[
y = -2x + 4
\]

Thus, the equation of the line that is perpendicular to the original line and shares the same y-intercept is:
\[
\text{D. } y = -2x + 4
\]