Question

graph the equation. then describe the transformation from the parent function f(x)=|x|.
( y = -\frac{1}{4}|x| )

choose the correct graph below.

a. graph a

b. graph b

c. graph c

d. graph d

Explanation:

Step1: Analyze the parent function

The parent function is \( f(x) = |x| \), which is a V - shaped graph with the vertex at the origin \((0,0)\), opening upwards, and has a slope of \( 1 \) for \( x\geq0 \) and \( - 1 \) for \( x < 0 \).

Step2: Analyze the transformation for \( y=-\frac{1}{4}|x| \)

• Reflection: The negative sign in front of the \( \frac{1}{4}|x| \) means the graph of \( y = |x| \) is reflected over the \( x \) - axis. So the graph will open downwards instead of upwards.
• Vertical Compression: The coefficient \( \frac{1}{4} \) (a value between \( 0 \) and \( 1 \)) causes a vertical compression of the graph of \( y = |x| \). For a function \( y = a|x| \), when \( 0<|a|<1 \), it is a vertical compression.

Now, let's analyze the options:

• Option A: The graph is a V - shaped graph opening downwards (due to the negative sign) and compressed vertically (since the slope is less steep than \( y = |x| \)) which matches the transformation of \( y=-\frac{1}{4}|x| \).
• Option B: The graph opens upwards, which does not match the reflection over the \( x \) - axis caused by the negative sign.
• Option C: The graph opens upwards, which does not match the reflection over the \( x \) - axis caused by the negative sign.
• Option D: The graph is a curve, not a V - shaped graph, so it does not represent an absolute - value function.

Answer:

A