Question

compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply.
graph of an absolute - value function with vertex at (3, 3) and y - intercept at (0, 6)
options:
□ a reflected over the x - axis
□ b horizontal shift left
□ c horizontal shift right
□ d vertical shift up
□ e vertical shift down
□ f graph is narrower
□ g graph is wider

Explanation:

Step1: Analyze the vertex of the parent function and the given graph

The parent function \( y = |x| \) has its vertex at \( (0, 0) \). The given graph has its vertex at \( (3, 3) \).

Step2: Analyze horizontal shift

To get from \( x = 0 \) (vertex of parent) to \( x = 3 \) (vertex of given graph), we shift 3 units to the right. So, horizontal shift right (option c) applies.

Step3: Analyze vertical shift

To get from \( y = 0 \) (vertex of parent) to \( y = 3 \) (vertex of given graph), we shift 3 units up. So, vertical shift up (option d) applies.

Step4: Analyze reflection

The graph of the parent function \( y = |x| \) opens upwards, and the given graph also opens upwards, so there is no reflection over the x - axis (option a is incorrect).

Step5: Analyze the width of the graph

The slope of the parent function \( y = |x| \) has a slope of \( \pm1 \). Let's find the slope of the given graph. From the vertex \( (3, 3) \) to \( (0, 6) \), the slope \( m=\frac{6 - 3}{0 - 3}=\frac{3}{-3}=- 1 \), and from \( (3, 3) \) to \( (6, 6) \), the slope \( m=\frac{6 - 3}{6 - 3}=\frac{3}{3}=1 \). The slopes are the same as the parent function, so the graph has the same width as the parent function (options f and g are incorrect).

Answer:

c. Horizontal Shift Right, d. Vertical Shift Up