Ask a New Question
Menu

Calculate the equation of a line that is perpendicular to the line shown on the graph and that also shares the same y-intercept.

(1 point)
Responses

y=2x−8
y equals 2 x minus 8

y=−12x+4
y equals negative Start Fraction 1 over 2 End Fraction x plus 4

y=−2x+4
y equals negative 2 x plus 4

y=12x−8
y equals Start Fraction 1 over 2 End Fraction x minus 8

1 answer

To find the equation of a line that is perpendicular to another line and shares the same y-intercept, follow these steps:

  1. Identify the slope of the original line: If the line's equation is in the form \(y = mx + b\), then \(m\) is the slope. Since we don't have the original line's equation in this scenario, you will need to determine it from the graph.

  2. Find the slope of the perpendicular line: If the slope of the original line is \(m\), the slope \(m'\) of the perpendicular line is given by: \[ m' = -\frac{1}{m} \]

  3. Use the shared y-intercept: Since the perpendicular line shares the same y-intercept (let's say this is \(b\)), the equation of the perpendicular line can be expressed as: \[ y = m'x + b \]

Now let's analyze the given responses:

  1. y = 2x - 8

    • Slope \(m = 2\) -> Perpendicular slope \(m' = -\frac{1}{2}\). With a y-intercept of \(-8\), this doesn't match.

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

    • Slope \(m = -\frac{1}{2}\) -> Perpendicular slope \(m' = 2\). With a y-intercept of \(4\), this doesn't seem consistent.

  3. y = -2x + 4

    • Slope \(m = -2\) -> Perpendicular slope \(m' = \frac{1}{2}\). With a y-intercept of \(4\), this is valid.

  4. y = \frac{1}{2}x - 8

    • Slope \(m = \frac{1}{2}\) -> Perpendicular slope \(m' = -2\). With a y-intercept of \(-8\), this does not match.

Based on this analysis, if the original line were to have a slope of \( -2\) and a y-intercept of \(4\), the correct choice is:

y = -2x + 4

Thus, the answer is:

y = -2x + 4.