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Part 1
Find an equation of the line. Write the equation using function notation.
Through left parenthesis 8 comma negative 5 right parenthesis​; perpendicular to 4yequalsxnegative 8
Question content area bottom
Part 1
The equation of the line is ​f(x)equals

enter your response here.

To find the equation of a line that is perpendicular to another line and passes through a given point, we follow these steps:

### Step 1: Find the slope of the given line
The given line equation is \( 4y = x - 8 \).

First, let's write it in the slope-intercept form \( y = mx + b \) to find its slope \( m \).

Divide both sides by 4:
\[
y = \frac{1}{4}x - 2
\]
So, the slope \( m_1 \) of the given line is \( \frac{1}{4} \).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope \( m_2 \) of the perpendicular line is:
\[
m_2 = -\frac{1}{\frac{1}{4}} = -4
\]

### Step 3: Use the point-slope form to write the equation
We use the point-slope form of a line equation \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point the line passes through and \( m \) is the slope.

The point given is \( (8, -5) \), so:
\[
y - (-5) = -4(x - 8)
\]
\[
y + 5 = -4x + 32
\]
Subtract 5 from both sides to get the equation in slope-intercept form:
\[
y = -4x + 27
\]

### Step 4: Write the equation using function notation
The equation in function notation is:
\[
f(x) = -4x + 27
\]

So, the equation of the line is \( f(x) = -4x + 27 \).

Would you like more details, or do you have any questions?

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1. How do you find the slope of a line from its equation?
2. What is the point-slope form of a line?
3. How do you calculate the negative reciprocal of a fraction?
4. What are the characteristics of perpendicular lines?
5. How do you convert a line equation into function notation?
6. What does the slope-intercept form of a line equation look like?
7. How can you determine if two lines are perpendicular?
8. Why do perpendicular lines have slopes that are negative reciprocals?

**Tip:** When given a line equation, converting it to slope-intercept form \( y = mx + b \) helps quickly identify the slope \( m \), which is essential for many line-related problems.

Learn how to find the equation of a line perpendicular to another line and passing through a given point (8, -5). This problem involves identifying slopes, using the point-slope form of a line equation, and understanding perpendicularity in lines. Suitable for Grades 9-12 students studying algebra and geometry.

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