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=−12x+4 y equals negative Start Fraction 1 over 2 End Fraction x plus 4 y=2x−8 y equals 2 x minus 8 y=−2x+4 y equals negative 2 x plus 4 y=12x−8
Answers
GPT-4o mini
To find the equation of a line that is perpendicular to a given line and has the same y-intercept, we first need to know the slope of the line from the graph. However, since I don't have access to the specific graph you're referring to, I'll guide you through how to derive a perpendicular line based on a given slope.
Assuming the slope of the original line is \( m \), the slope of any line that is perpendicular to it would be \( -\frac{1}{m} \).
Now, let's consider the y-intercept of the line is represented as \( b \). Therefore, the equation of the new line would be:
\[
y = -\frac{1}{m}x + b
\]
Now, let's analyze the options given and see if we can find any that share a common y-intercept and have slopes that are negatives reciprocals.
1. \( y = -12x + 4 \) (Slope = -12)
2. \( y = -\frac{1}{2}x + 4 \) (Slope = -\frac{1}{2})
3. \( y = 2x - 8 \) (Slope = 2)
4. \( y = 2x - 8 \) (Slope = 2)
5. \( y = -2x + 4 \) (Slope = -2)
6. \( y = 12x - 8 \) (Slope = 12)
To derive possible slopes that would be perpendicular, we must apply the negative reciprocal formula:
- If \( m = -12 \), then \( -\frac{1}{m} = \frac{1}{12} \)
- If \( m = -\frac{1}{2} \), then \( -\frac{1}{m} = 2 \)
- If \( m = 2 \), then \( -\frac{1}{m} = -\frac{1}{2} \)
- If \( m = -2 \), then \( -\frac{1}{m} = \frac{1}{2} \)
- If \( m = 12 \), then \( -\frac{1}{m} = -\frac{1}{12} \)
Next, we also check for the y-intercepts in the equations provided:
- The equations \( y = -12x + 4 \) and \( y = -2x + 4 \) have y-intercept 4.
- The equations \( y = 2x - 8 \) and its duplicate \( y = 2x - 8 \) have y-intercept -8.
- The equations \( y = -\frac{1}{2}x + 4\) and \( y = 12x - 8\) both yield different intercepts.
From this, we compare their slopes:
If we take the equations with \( 4 \) as the y-intercept:
- \( y = -12x + 4 \) (slope = -12)
- \( y = -2x + 4 \) (slope = -2)
The line that is perpendicular to a slope of \( -2 \) is \( \frac{1}{2} \), thus confirming that it shares the y-intercept of \( 4 \).
From the above options, the line that is perpendicular and shares the y-intercept of 4 is:
**y = -2x + 4**
Assuming the slope of the original line is \( m \), the slope of any line that is perpendicular to it would be \( -\frac{1}{m} \).
Now, let's consider the y-intercept of the line is represented as \( b \). Therefore, the equation of the new line would be:
\[
y = -\frac{1}{m}x + b
\]
Now, let's analyze the options given and see if we can find any that share a common y-intercept and have slopes that are negatives reciprocals.
1. \( y = -12x + 4 \) (Slope = -12)
2. \( y = -\frac{1}{2}x + 4 \) (Slope = -\frac{1}{2})
3. \( y = 2x - 8 \) (Slope = 2)
4. \( y = 2x - 8 \) (Slope = 2)
5. \( y = -2x + 4 \) (Slope = -2)
6. \( y = 12x - 8 \) (Slope = 12)
To derive possible slopes that would be perpendicular, we must apply the negative reciprocal formula:
- If \( m = -12 \), then \( -\frac{1}{m} = \frac{1}{12} \)
- If \( m = -\frac{1}{2} \), then \( -\frac{1}{m} = 2 \)
- If \( m = 2 \), then \( -\frac{1}{m} = -\frac{1}{2} \)
- If \( m = -2 \), then \( -\frac{1}{m} = \frac{1}{2} \)
- If \( m = 12 \), then \( -\frac{1}{m} = -\frac{1}{12} \)
Next, we also check for the y-intercepts in the equations provided:
- The equations \( y = -12x + 4 \) and \( y = -2x + 4 \) have y-intercept 4.
- The equations \( y = 2x - 8 \) and its duplicate \( y = 2x - 8 \) have y-intercept -8.
- The equations \( y = -\frac{1}{2}x + 4\) and \( y = 12x - 8\) both yield different intercepts.
From this, we compare their slopes:
If we take the equations with \( 4 \) as the y-intercept:
- \( y = -12x + 4 \) (slope = -12)
- \( y = -2x + 4 \) (slope = -2)
The line that is perpendicular to a slope of \( -2 \) is \( \frac{1}{2} \), thus confirming that it shares the y-intercept of \( 4 \).
From the above options, the line that is perpendicular and shares the y-intercept of 4 is:
**y = -2x + 4**