Question

select the correct answer.
the graph of the rational function f is shown.
which graph is the graph of g(x) = -f(x)?

Explanation:

Step1: Recall Reflection Over x - axis

The transformation \(g(x)=-f(x)\) represents a reflection of the graph of \(y = f(x)\) over the \(x\) - axis. For any point \((x,y)\) on the graph of \(f(x)\), the corresponding point on the graph of \(g(x)\) will be \((x, - y)\).

Step2: Analyze Key Features of \(f(x)\)

• The vertical asymptote of \(f(x)\) is at \(x=-1\) (the dashed vertical line). The vertical asymptote will remain the same for \(g(x)\) because the transformation \(y=-f(x)\) does not affect the vertical asymptote (vertical asymptotes are determined by the denominator of the rational function, and this transformation is a reflection over the \(x\) - axis, not a horizontal shift or change in the denominator).
• The horizontal asymptote of \(f(x)\) is \(y = 2\) (the dashed horizontal line). After reflection over the \(x\) - axis, the horizontal asymptote of \(g(x)\) will be \(y=-2\) (since we reflect the \(y\) - values over the \(x\) - axis).
• The graph of \(f(x)\) passes through the origin \((0,0)\). After reflection, the point \((0,0)\) will still be on the graph of \(g(x)\) because \(-0 = 0\).
• For the left - hand branch (where \(x\lt - 1\)) of \(f(x)\), the \(y\) - values are positive (above the \(x\) - axis). After reflection, the \(y\) - values of the left - hand branch of \(g(x)\) will be negative (below the \(x\) - axis). For the right - hand branch (where \(x\gt - 1\)) of \(f(x)\), the \(y\) - values are positive (above the \(x\) - axis). After reflection, the \(y\) - values of the right - hand branch of \(g(x)\) will be negative (below the \(x\) - axis).

Step3: Analyze Option A

• In option A, the vertical asymptote is still at \(x = - 1\) (correct, as the vertical asymptote should not change).
• The horizontal asymptote of the graph in option A appears to be \(y=-2\) (which is the reflection of \(y = 2\) over the \(x\) - axis, correct).
• The left - hand branch of the graph in option A is below the \(x\) - axis (consistent with the reflection of the left - hand branch of \(f(x)\) which was above the \(x\) - axis) and the right - hand branch is also below the \(x\) - axis (consistent with the reflection of the right - hand branch of \(f(x)\) which was above the \(x\) - axis). The point \((0,0)\) is still on the graph (since \(-0 = 0\)).

Answer:

A