Question

the absolute - value function (f) has a range of ((-5,infty)), and a graph that is symmetric about the line (x = 4), and has a (y) - intercept of 1. identify the graph of (f).

Explanation:

Step1: Analyze range

The range of the function is $(- 5,\infty)$. This means the minimum - value of the function is $y=-5$.

Step2: Analyze axis of symmetry

The axis of symmetry is $x = 4$. So the vertex of the absolute - value function lies on the line $x = 4$.

Step3: Analyze y - intercept

The y - intercept is 1, which means the function passes through the point $(0,1)$.

We look for a graph that has a vertex with a y - value of $-5$ on the line $x = 4$ and passes through $(0,1)$.

Answer:

The graph that meets these criteria (a vertex at $(4,-5)$ and y - intercept of 1) is the correct one among the given options. Without the clear - view of the options due to the blurriness of the image, the general approach to identify the graph is by checking for a V - shaped graph (characteristic of absolute - value functions) with vertex at $(4,-5)$ and passing through $(0,1)$.