Question

write an equation for graphed function by using transformations of the graphs of one of the basic functions.

Explanation:

Step1: Identify basic parent function

The graph is a V-shaped piecewise linear function, so the parent function is $f(x)=|x|$, which has its vertex at $(0,0)$.

Step2: Identify vertex shift

The vertex of the given graph is at $(1,4)$. This means the parent function is shifted 1 unit right and 4 units up. The transformed function at this stage is $y=|x-1|+4$.

Step3: Identify slope/vertical stretch

For the parent function $|x|$, the slope of the right side is 1. For the given graph, from the vertex $(1,4)$ to $(0,3)$, the slope is $\frac{3-4}{0-1}=1$; from $(1,4)$ to $(3,2)$, the slope is $\frac{2-4}{3-1}=-1$. Wait, no—checking the left side: from $(-3,0)$ to $(0,3)$, slope is $\frac{3-0}{0-(-3)}=1$, which matches the right side slope magnitude. But wait, the parent function $|x|$ passes through $(1,1)$, but our function passes through $(1,4)$ (vertex), $(0,3)$ which is $|0-1|+3=1+3=4$? No, correct adjustment: the parent function $|x|$ has a y-intercept at 0, our function has y-intercept at 3. When $x=0$, $y=3$. Using the vertex form $y=a|x-h|+k$, where $(h,k)=(1,4)$:
Substitute $x=0, y=3$:
$3=a|0-1|+4$
$3=a(1)+4$
$a=3-4=-1$

Step4: Verify the function

Test $x=-3$: $y=-1|-3-1|+4=-1(4)+4=0$, which matches the graph.
Test $x=3$: $y=-1|3-1|+4=-1(2)+4=2$, which matches the graph.

Answer:

$y = -|x-1| + 4$