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+4
y equals negative 2 x plus 4
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=12x−8
(1 point)
Responses
y=−2x+4
y equals negative 2 x plus 4
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=12x−8
Answers
GPT-4o mini
To find the equation of a line that is perpendicular to a given line and shares the same y-intercept, we need to follow these steps:
1. **Determine the slope of the original line.** If we don't have the original line's equation, we can infer one from the problem; however, usually, if you look at a graph, the slope can be determined visually.
2. **Find the slope of the perpendicular line.** The slopes of perpendicular lines are negative reciprocals of each other. If the slope of the original line is \( m \), then the slope of the perpendicular line is \( -\frac{1}{m} \).
3. **Use the same y-intercept.** The y-intercept is typically given in the form of \( y = mx + b \), where \( b \) is the y-intercept.
From the responses you have listed, it appears the y-intercept we will use is 4 (since all options share this intercept).
Assuming the original line's slope guesses from your choices: if it were to be \( -2 \) (for example), the negative reciprocal would be \( \frac{1}{2} \).
Now let's analyze the provided options based on extractable patterns:
1. **Option 1:** \( y = -2x + 4 \) → Slope = -2 (not perpendicular if original line is also -2)
2. **Option 2:** \( y = 2x - 8 \) → Slope = 2 (not matching)
3. **Option 3:** \( y = -\frac{1}{2}x + 4 \) → Slope = -1/2 (also not matching)
4. **Option 4:** \( y = 12x - 8 \) → Slope = 12 (too steep)
We need a line with a slope that is the negative reciprocal of the original slope and also has a y-intercept of 4.
Assuming the original line had a slope of -2, then its perpendicular slope would be 1/2, leading us to take the form:
### Step: Write the Equation
So the equation becomes:
\[
y = \frac{1}{2}x + 4
\]
However, since none of the choices lists this option directly, **without seeing the graph that denotes the slope we are considering**, it appears at least some coefficients are mismatched.
To conclude, we would need to know the original slope (of the line in the graph) directly to identify the perpendicular slope. If you can evaluate the choice options given a full equation that directly corresponds to a line on the graph, we could pinpoint further which option is correct based on y-intercept confirmation and perpendicular slope requirement.
1. **Determine the slope of the original line.** If we don't have the original line's equation, we can infer one from the problem; however, usually, if you look at a graph, the slope can be determined visually.
2. **Find the slope of the perpendicular line.** The slopes of perpendicular lines are negative reciprocals of each other. If the slope of the original line is \( m \), then the slope of the perpendicular line is \( -\frac{1}{m} \).
3. **Use the same y-intercept.** The y-intercept is typically given in the form of \( y = mx + b \), where \( b \) is the y-intercept.
From the responses you have listed, it appears the y-intercept we will use is 4 (since all options share this intercept).
Assuming the original line's slope guesses from your choices: if it were to be \( -2 \) (for example), the negative reciprocal would be \( \frac{1}{2} \).
Now let's analyze the provided options based on extractable patterns:
1. **Option 1:** \( y = -2x + 4 \) → Slope = -2 (not perpendicular if original line is also -2)
2. **Option 2:** \( y = 2x - 8 \) → Slope = 2 (not matching)
3. **Option 3:** \( y = -\frac{1}{2}x + 4 \) → Slope = -1/2 (also not matching)
4. **Option 4:** \( y = 12x - 8 \) → Slope = 12 (too steep)
We need a line with a slope that is the negative reciprocal of the original slope and also has a y-intercept of 4.
Assuming the original line had a slope of -2, then its perpendicular slope would be 1/2, leading us to take the form:
### Step: Write the Equation
So the equation becomes:
\[
y = \frac{1}{2}x + 4
\]
However, since none of the choices lists this option directly, **without seeing the graph that denotes the slope we are considering**, it appears at least some coefficients are mismatched.
To conclude, we would need to know the original slope (of the line in the graph) directly to identify the perpendicular slope. If you can evaluate the choice options given a full equation that directly corresponds to a line on the graph, we could pinpoint further which option is correct based on y-intercept confirmation and perpendicular slope requirement.