Question

1. explain the transformation from the parent function ( f(x) = |x| ) to ( f(x) = \frac{4}{5}|x - 4| + 5 ). you have to distinguish between vertical/horizontal expansions/compressions. (8 pts)

a.

Explanation:

Step1: Recall Parent Function

The parent function is \( f(x) = |x| \), which is a V - shaped graph with vertex at \((0,0)\).

Step2: Analyze Horizontal Shift

For the function \( y = |x - h| \), the graph of \( y = |x| \) is shifted \( h \) units horizontally. In \( f(x)=\frac{4}{5}|x - 4|+5 \), the \( x-4 \) inside the absolute value means the graph of \( y = |x| \) is shifted 4 units to the right (since \( h = 4>0 \)).

Step3: Analyze Vertical Compression/Stretch

For the function \( y = a|x| \), if \( 0 < |a|<1 \), it is a vertical compression, and if \( |a|>1 \), it is a vertical stretch. Here, \( a=\frac{4}{5} \), and since \( 0<\frac{4}{5}<1 \), the graph of \( y = |x - 4| \) (after horizontal shift) is vertically compressed by a factor of \( \frac{4}{5} \).

Step4: Analyze Vertical Shift

For the function \( y = |x|+k \), the graph is shifted \( k \) units vertically. Here, \( k = 5>0 \), so the graph of \( y=\frac{4}{5}|x - 4| \) (after horizontal shift and vertical compression) is shifted 5 units up.

Answer:

1. Horizontal Shift: The graph of \( f(x)=|x| \) is shifted 4 units to the right to get \( y = |x - 4| \).
2. Vertical Compression: The graph of \( y = |x - 4| \) is vertically compressed by a factor of \( \frac{4}{5} \) to get \( y=\frac{4}{5}|x - 4| \) (since \( 0<\frac{4}{5}<1 \), it is a compression).
3. Vertical Shift: The graph of \( y=\frac{4}{5}|x - 4| \) is shifted 5 units up to get \( f(x)=\frac{4}{5}|x - 4|+5 \).