To find the graph of \(-f(x)\) given \(f(x) = (x - 4)^2 + 2\), we first need to understand what \(-f(x)\) represents.
Starting with the function:
\[ f(x) = (x - 4)^2 + 2 \]
Now, to find \(-f(x)\):
\[ -f(x) = -[(x - 4)^2 + 2] = -(x - 4)^2 - 2 \]
This means that the graph of \(-f(x)\) is obtained by taking the original graph of \(f(x)\) and reflecting it across the x-axis, and then shifting it down by 2 units.
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Reflection:
- The graph of \(f(x)\) opens upwards since it is a quadratic function with a positive leading coefficient. Reflecting it across the x-axis will make it open downwards.
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Shift Down:
- The entire graph of the reflected function is then shifted down by 2.
Key Points about the Graph of \(-f(x)\):
- The vertex of the original function \(f(x)\) is at \((4, 2)\).
- Reflecting the vertex \((4, 2)\) across the x-axis gives \((4, -2)\).
- Shifting this point down by 2 yields the new vertex at \((4, -2 - 2) = (4, -4)\).
- The graph will open downwards from this vertex.
If you have specific options to choose from, the graph you are looking for will be a downward-opening parabola with its vertex at \((4, -4)\). It will also be symmetrical about the vertical line \(x = 4\).
Make sure to look for these characteristics when examining the options for the graph of \(-f(x)\).