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.
(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 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
Answers
GPT-4o mini
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**.
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**.