To determine the graph of the function [tex]\( h(x) = -(x - 2)^2 \)[/tex], we can use transformations of the graph of the basic quadratic function [tex]\( f(x) = x^2 \)[/tex]. Here's a detailed, step-by-step explanation of the transformations:
1. Start with the basic quadratic function [tex]\( f(x) = x^2 \)[/tex].
- The graph of [tex]\( f(x) = x^2 \)[/tex] is a parabola that opens upwards with its vertex at the origin, [tex]\((0, 0)\)[/tex].
2. Apply a horizontal shift.
- The expression [tex]\( (x - 2) \)[/tex] suggests a horizontal shift. Specifically, [tex]\( (x - 2) \)[/tex] means the graph of [tex]\( f(x) = x^2 \)[/tex] is shifted to the right by 2 units.
- This transformation moves the vertex of the parabola from [tex]\((0, 0)\)[/tex] to [tex]\((2, 0)\)[/tex].
- After this horizontal shift, the transformed function is [tex]\( g(x) = (x - 2)^2 \)[/tex].
3. Apply a reflection across the x-axis.
- The negative sign in front of [tex]\( (x - 2)^2 \)[/tex] indicates a reflection across the x-axis.
- Reflecting the graph of [tex]\( g(x) = (x - 2)^2 \)[/tex] across the x-axis inverts it. This means every point [tex]\((x, y)\)[/tex] on the parabola [tex]\( g(x) = (x - 2)^2 \)[/tex] is transformed to the point [tex]\((x, -y)\)[/tex] on the new parabola.
- Consequently, the vertex of the parabola [tex]\( g(x) = (x - 2)^2 \)[/tex], which is at [tex]\((2, 0)\)[/tex], remains at [tex]\((2, 0)\)[/tex] because it lies on the x-axis, but the parabola now opens downwards.
After applying these transformations, we get the final function:
[tex]\[ h(x) = -(x - 2)^2 \][/tex]
From this transformation process, the graph of [tex]\( h(x) \)[/tex]:
- Is shifted 2 units to the right from the original graph of [tex]\( f(x) = x^2 \)[/tex],
- Is reflected across the x-axis, resulting in a downward-opening parabola.
Therefore, the graph of [tex]\( h(x) = -(x - 2)^2 \)[/tex] is obtained by shifting the graph of [tex]\( f(x) = x^2 \)[/tex] 2 units to the right and reflecting it across the x-axis.
