Question

1. (+4 points)

8 3 8
the equation of line a, is y = ¼ x
a. which of the following equations
represent line h? select all that apply.
□ y = ¼ x -5
□ y = ¼ x + 5
□ ¼ x - 5 = y
□ y = -5 + ¼ x
□ y = 5 - ¼ x
b. if line h is an image of line a, which type of transformation do their equations demonstrate? rotation,
reflection, translation, or dilation note: be specific with the description of your transformation

Explanation:

Part (a)

Step1: Analyze Line a and Line h

Line \( a \) has equation \( y = \frac{1}{4}x \). Line \( h \) is parallel to line \( a \) (same slope \( \frac{1}{4} \)) and shifted down. The slope-intercept form is \( y = mx + b \), where \( m \) is slope and \( b \) is y-intercept. Line \( h \) has \( m = \frac{1}{4} \) and \( b = -5 \).

Step2: Check Each Equation

• \( y = \frac{1}{4}x - 5 \): Slope \( \frac{1}{4} \), y-intercept -5. Correct.
• \( y = \frac{1}{4}x + 5 \): Y-intercept +5 (shift up). Incorrect.
• \( \frac{1}{4}x - 5 = y \): Same as \( y = \frac{1}{4}x - 5 \). Correct.
• \( y = -5 + \frac{1}{4}x \): Rearranged \( y = \frac{1}{4}x - 5 \). Correct.
• \( y = 5 - \frac{1}{4}x \): Slope \( -\frac{1}{4} \). Incorrect.

Line \( a \) ( \( y = \frac{1}{4}x \) ) and line \( h \) ( \( y = \frac{1}{4}x - 5 \) ) have the same slope (parallel) but different y-intercepts. This means line \( h \) is a vertical shift (translation) down 5 units from line \( a \). Rotation would change slope, reflection would flip slope sign, dilation changes slope magnitude. Here, only position changes (translation).

Answer:

(Part a):
\( y = \frac{1}{4}x - 5 \), \( \frac{1}{4}x - 5 = y \), \( y = -5 + \frac{1}{4}x \)

Part (b)