Question

question 16 (1 point)
compare the equation to the absolute value parent function ($y = |x|$). select all the transformations that apply.
$y = \frac{1}{5}|x| + 3$
\\(\square\\) a reflected over the x - axis
\\(\square\\) b horizontal shift left
\\(\square\\) c horizontal shift right
\\(\square\\) d vertical shift up
\\(\square\\) e vertical shift down
\\(\square\\) f graph is narrower
\\(\square\\) g graph is wider

Explanation:

Step1: Analyze vertical shift

The parent function is \( y = |x| \), and the given function is \( y=\frac{1}{5}|x| + 3 \). The \( + 3 \) at the end indicates a vertical shift. In the form \( y=a|x - h|+k \), \( k \) is the vertical shift. Here \( k = 3>0 \), so it's a vertical shift up.

Step2: Analyze vertical stretch/compression

The coefficient of \( |x| \) is \( \frac{1}{5} \), where \( 0<\frac{1}{5}<1 \). When the coefficient \( a \) of \( |x| \) satisfies \( 0 < |a|<1 \), the graph is wider (vertically compressed) compared to the parent function \( y = |x| \) (where \( a = 1 \)).

Step3: Check reflection and horizontal shifts

There is no negative sign in front of \( |x| \), so no reflection over the x - axis. Also, there is no \( h \) value (the term inside the absolute value is just \( |x| \), not \( |x - h| \) with \( h
eq0 \)), so there are no horizontal shifts (left or right).

Answer:

d. Vertical Shift Up, g. Graph is Wider