Question

here is the graph that represents a function which equation represents the function? f(x)=|x|−5 h(x)=|x+5| j(x)=|x|+5 g(x)= |x−5|

Explanation:

Step1: Analyze the vertex of the graph

The graph of an absolute - value function \(y = |x - h|+k\) has its vertex at \((h,k)\). From the given graph, the vertex is at \((0, - 5)\). So \(h = 0\) and \(k=-5\).

Step2: Recall the general form of absolute - value function

The general form of an absolute - value function is \(y=a|x - h|+k\). Since the vertex is \((0,-5)\), the function has the form \(y = a|x| - 5\). Also, the graph opens upwards, so \(a = 1\) (because for \(y = |x|\), the graph opens upwards with a slope of \(1\) for \(x\geq0\) and \(- 1\) for \(x<0\), and our graph has the same shape as \(y = |x|\) but shifted down by \(5\) units).

Let's check the options:

• For \(f(x)=|x|-5\): When \(x = 0\), \(f(0)=|0|-5=-5\), which matches the vertex \((0, - 5)\).
• For \(h(x)=|x + 5|\): The vertex of \(y=|x + 5|\) is at \((-5,0)\), which does not match the vertex of the given graph.
• For \(j(x)=|x|+5\): When \(x = 0\), \(j(0)=|0| + 5=5\), the vertex would be at \((0,5)\), which does not match.
• For \(g(x)=|x - 5|\): The vertex of \(y = |x - 5|\) is at \((5,0)\), which does not match the vertex of the given graph.

Answer:

\(f(x)=|x|-5\)