Question

1. a teenager sets a goal to earn $75 each week from an after - school job and uses the equation y = |x - 75| to find the difference, in dollars, between the teenager’s actual earnings, x, and goal earnings from the after - school job. which graph best represents the equation?

a
b
c
d

Explanation:

Step1: Recall Absolute Value Graph

The equation \( y = |x - 75| \) is an absolute - value function. The general form of an absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex of the V - shaped graph. For \( y = |x - 75| \), \( a = 1\), \( h=75\), and \( k = 0\). So the vertex of the graph is at the point \((75,0)\).

Step2: Analyze the Behavior of the Function

• When \( x<75\), \( y=-(x - 75)=-x + 75\). This is a linear function with a slope of \(- 1\).
• When \( x\geq75\), \( y=x - 75\). This is a linear function with a slope of \(1\).

Step3: Analyze the Graphs

• For graph A: The vertex is at \((40,0)\), which does not match the vertex \((75,0)\) of \(y = |x - 75|\).
• For graph B: The graph is not a V - shaped graph with vertex at \((75,0)\), and its shape is not consistent with the absolute - value function \(y = |x - 75|\).
• For graph C: The vertex is not at \((75,0)\), so it is not the graph of \(y = |x - 75|\).
• For graph D: The vertex is at \((75,0)\). When \(x<75\), the slope of the left - hand side of the V is \(- 1\) (as \(y=-x + 75\)), and when \(x\geq75\), the slope of the right - hand side of the V is \(1\) (as \(y=x - 75\)), which is consistent with the function \(y = |x - 75|\).

Answer:

The graph labeled D.