write an equation of the absolute value function graph.

Explanation:

Step1: Recall absolute - value function form

The general form of an absolute - value function is $y=a|x - h|+k$, where $(h,k)$ is the vertex of the graph. The vertex of the given graph is $(0,4)$, so $h = 0$ and $k = 4$. Then the function is $y=a|x|+4$.

Step2: Find the value of a

Substitute the point $(4,3)$ into the equation $y=a|x|+4$. We get $3=a|4|+4$. Simplify the equation: $3 = 4a+4$. Subtract 4 from both sides: $3 - 4=4a$, so $-1 = 4a$. Solve for $a$: $a=-\frac{1}{4}$.

Answer:

$y =-\frac{1}{4}|x|+4$