Question

which of the following equations represents the graph pictured?
f(x) = -|x| + 1
f(x) = |-x - 1|
f(x) = |-x + 1| - 1
f(x) = -|x + 1|

Explanation:

Step1: Analyze the vertex of the graph

The graph of an absolute - value function \(y = a|x - h|+k\) has its vertex at \((h,k)\). From the given graph, we can see that the vertex is at \((- 1,0)\).

Step2: Analyze the direction of the graph

The graph opens downwards, which means that the coefficient of the absolute - value term \(a\) is negative.

Step3: Analyze each option

• Option 1: For \(f(x)=-|x| + 1\), the vertex is at \((0,1)\), which does not match the vertex of the given graph \((-1,0)\).
• Option 2: For \(f(x)=|-x - 1|=|-(x + 1)|=|x + 1|\), the coefficient of the absolute - value term is positive (\(a = 1\)), so the graph opens upwards, which does not match the direction of the given graph (which opens downwards).
• Option 3: For \(f(x)=|-x + 1|-1=|-(x - 1)|-1=|x - 1|-1\), the vertex is at \((1,-1)\), which does not match the vertex of the given graph \((-1,0)\).
• Option 4: For \(f(x)=-|x + 1|\), the vertex form of the absolute - value function is \(y=-|x-(-1)|+0\), so the vertex is at \((-1,0)\) and the coefficient \(a=-1\) (negative), so the graph opens downwards, which matches the characteristics of the given graph.

Answer:

\(f(x)=-|x + 1|\) (the fourth option)