Write the​ slope-intercept equation of the function f whose graph passes through (-4,6) and is perpendicular to the line that has an​ x-intercept of 2 and a​ y-intercept of -4 .

To find the slope-intercept form of the function \( f(x) \), we need to follow these steps:

### Step 1: Determine the slope of the given line.
The given line has an x-intercept of 2 and a y-intercept of -4. 

To find the slope (\(m\)) of this line, use the slope formula between the two points \((x_1, y_1)\) and \((x_2, y_2)\):
- \( (x_1, y_1) = (2, 0) \)
- \( (x_2, y_2) = (0, -4) \)

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{0 - 2} = \frac{-4}{-2} = 2
\]

So, the slope of the given line is \(2\).

### Step 2: Find the slope of the line perpendicular to the given line.
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.

\[
m_{\text{perpendicular}} = -\frac{1}{2}
\]

### Step 3: Use the point-slope form to find the equation of the perpendicular line.
We have a point \((-4, 6)\) and a slope of \(-\frac{1}{2}\).

The point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
Substitute the values:
\[
y - 6 = -\frac{1}{2}(x + 4)
\]

### Step 4: Convert to slope-intercept form (\(y = mx + b\)).
Expand and simplify:
\[
y - 6 = -\frac{1}{2}x - 2
\]
\[
y = -\frac{1}{2}x + 4
\]

So, the slope-intercept equation of the function \( f(x) \) is:
\[
f(x) = -\frac{1}{2}x + 4
\]

Do you want details on any step or have any questions?

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Here are some related questions you might find interesting:

1. How do you find the slope of a line given two points?
2. What is the significance of the negative reciprocal in perpendicular slopes?
3. How can you verify that two lines are perpendicular?
4. How do you find the equation of a line parallel to a given line?
5. What is the general form of a linear equation, and how does it relate to the slope-intercept form?

**Tip:** Remember, when working with perpendicular lines, always take the negative reciprocal of the original slope to find the slope of the perpendicular line.

Write the slope-intercept equation of the function f whose graph passes through (-4,6) and is perpendicular to the line that has an x-intercept of 2 and a y-intercept of -4.

Learn how to find the slope-intercept equation of a function passing through the point (-4,6) that is perpendicular to a line with given intercepts. This problem involves key algebraic concepts, including slope calculations and the slope-intercept form, suitable for students in Grades 8-10.

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