Question

which of the following linear functions is perpendicular to the given linear function?
\\y = 4x + 4\\

a. \\(y = 4x - \frac{1}{4}\\)

b. \\(2y = 8x - 16\\)

c. \\(y = -\frac{1}{4}x - 8\\)

d. \\(y = \frac{1}{4}x - 8\\)

Explanation:

🆕 New Concept Discovered: Slopes of Perpendicular Lines
Opposite reciprocal slopes cross at right angles.

Step 1: Identify the slope of the given line

The given linear function is:

$$ y = 4x + 4 $$

This equation is written in slope-intercept form, \( y = mx + b \), where \( m \) represents the slope.

The slope of the given line is:

$$ m_1 = 4 $$

Step 2: Determine the perpendicular slope

For two lines to be perpendicular, their slopes must be negative reciprocals (opposite reciprocals) of each other. This means their product is \(-1\):

$$ m_1 \cdot m_2 = -1 $$

To find the perpendicular slope \( m_2 \), we take the reciprocal of \( 4 \) and change its sign:

$$ m_2 = -\frac{1}{4} $$

Step 3: Match with the options

We look for the option that has a slope of \( -\frac{1}{4} \):

• A: \( y = 4x - \frac{1}{4} \) (Slope is \( 4 \))
• B: \( 2y = 8x - 16 \implies y = 4x - 8 \) (Slope is \( 4 \))
• C: \( y = -\frac{1}{4}x - 8 \) (Slope is \( -\frac{1}{4} \))
• D: \( y = \frac{1}{4}x - 8 \) (Slope is \( \frac{1}{4} \))

Option C has the correct perpendicular slope of \( -\frac{1}{4} \).

Answer:

C