write a possible equation for the following graph using transformations.

Explanation:

Step1: Identify the parent - function

The parent - function of an absolute - value function is $y = |x|$.

Step2: Determine the vertex

The vertex of the given graph is at $(4,0)$. For the parent - function $y = |x|$ with vertex $(0,0)$, a horizontal shift of $h$ units to the right is given by $y=|x - h|$. Here, $h = 4$, so we have $y=|x - 4|$.

Step3: Determine the vertical stretch or shrink and vertical shift

The $y$ - intercept of the graph is at $(0,4)$. Substitute $x = 0$ into $y=a|x - 4|+k$. Since the vertex is at $(4,0)$ so $k = 0$. When $x = 0$, $y=a|0 - 4|+0$. Given $y = 4$ when $x = 0$, we have $4=a| - 4|$, so $4 = 4a$, and $a = 1$.

Answer:

$y=|x - 4|$