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+4 y equals negative 2 x plus 4 y=2x−8 y equals 2 x minus 8 y=12x−8

Answer 1

Are you sure? The two coordinates are, (-8,0) and (0,4)

Answer 2

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

1. Determine the slope of the original line.
2. Calculate the slope of the perpendicular line.
3. Use the shared y-intercept to write the equation of the perpendicular line.

### Step 1: Determine the slope of the original line

If we assume that the given line has a general form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Let's say the line from the graph has a slope \(m\).

### Step 2: Calculate the slope of the perpendicular line

The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. If the original line has a slope of \(m\), then the perpendicular slope \(m_{\perp}\) is:

\[
m_{\perp} = -\frac{1}{m}
\]

### Step 3: Write the equation of the perpendicular line

Since both lines share the same y-intercept, let's say the y-intercept is \(b\). Thus, the equation of the perpendicular line can be written as:

\[
y = m_{\perp}x + b
\]

Now, you provided several options:

- \(y = -12x + 4\)
- \(y = -\frac{1}{2}x + 4\)
- \(y = -2x + 4\)
- \(y = 2x - 8\)
- \(y = 12x - 8\)

### Choosing the correct options

Assuming we have determined the slope \(m\) for the original line to be among \(-12\), \(-\frac{1}{2}\), \(-2\), \(2\), and \(12\), we can calculate the negative reciprocal for each:

1. Slope \(m = -12 \Rightarrow m_{\perp} = \frac{1}{12}\) (not present)
2. Slope \(m = -\frac{1}{2} \Rightarrow m_{\perp} = 2\) (present in \(y = 2x - 8\))
3. Slope \(m = -2 \Rightarrow m_{\perp} = \frac{1}{2}\) (not present)
4. Slope \(m = 2 \Rightarrow m_{\perp} = -\frac{1}{2}\) (present in \(y = -\frac{1}{2}x + 4\))
5. Slope \(m = 12 \Rightarrow m_{\perp} = -\frac{1}{12}\) (not present)

### Final Result

Since the options that maintain the same y-intercept \(b = 4\) are \(y = -\frac{1}{2}x + 4\) and \(y = 2x - 8\), if \(b = 4\) for the perpendicular line:

The correct answer that shares the y-intercept of 4 and has a slope of \(2\) is:

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

Answer 3

Thank you for providing the coordinates. Now we can determine the slope of the line passing through the points \((-8, 0)\) and \((0, 4)\).

### Step 1: Calculate the slope of the original line

To find the slope \(m\) of the line passing through the points \((x_1, y_1) = (-8, 0)\) and \((x_2, y_2) = (0, 4)\), we can use the formula for slope:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the coordinates:

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

### Step 2: Determine the slope of the perpendicular line

The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, if the slope of the original line is \(\frac{1}{2}\), the slope of the perpendicular line \(m_{\perp}\) is:

\[
m_{\perp} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]

### Step 3: Use the shared y-intercept to write the equation of the perpendicular line

Since the original line has a y-intercept of \(4\) (from the point \((0, 4)\)), the equation of the line that is perpendicular and shares the same y-intercept will be:

\[
y = -2x + 4
\]

### Final Answer

Thus, the equation of the line that is perpendicular to the original line and shares the same y-intercept is:

**\(y = -2x + 4\)**.

This matches one of the provided options.