Question

function 1 and function 2 are both linear functions. the equation for function 1 is ( y = -\frac{4}{3}x + 2 ). the points ( (0, -8) ), ( (4, -5) ), and ( (8, -2) ) lie on function 2. which statement is correct?
a. the slopes of the lines for the functions are equal and the lines are perpendicular.
b. the slopes of the lines for the functions are negative reciprocals and the lines are perpendicular.
c. the slopes of the lines for the functions are equal and the lines are parallel.
d. the slope of the line for function 1 is greater than the slope of the line for function 2.

Explanation:

Step1: Find slope of Function 1

Function 1: \( y = -\frac{4}{3}x + 2 \), slope \( m_1 = -\frac{4}{3} \)

Step2: Calculate slope of Function 2

Using points \((0, -8)\), \((4, -5)\), \((8, -2)\). Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
For \((0, -8)\) and \((4, -5)\): \( m_2 = \frac{-5 - (-8)}{4 - 0} = \frac{3}{4} \)

Step3: Analyze slope relationship

Check \( m_1 \times m_2 = -\frac{4}{3} \times \frac{3}{4} = -1 \). When slopes multiply to -1, lines are perpendicular. Also, \( m_2 \) is negative reciprocal of \( m_1 \) (since \( -\frac{4}{3} \) and \( \frac{3}{4} \) are negative reciprocals: \( -\frac{4}{3} = -\frac{1}{\frac{3}{4}} \) reversed sign and reciprocal).

Step4: Evaluate options

• Option A: Slopes not equal (\( -\frac{4}{3}

eq \frac{3}{4} \)), so A wrong.

• Option B: Slopes are negative reciprocals (as shown) and lines perpendicular (product -1), so B correct.
• Option C: Slopes not equal, not parallel, C wrong.
• Option D: \( -\frac{4}{3} \approx -1.33 \), \( \frac{3}{4} = 0.75 \). \( -1.33 < 0.75 \), so D wrong.

Answer:

B. The slopes of the lines for the functions are negative reciprocals and the lines are perpendicular.