Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -(x − 2)²

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Start by recalling the graph of the standard quadratic function \(f(x) = x^{2}\). This is a parabola opening upwards with its vertex at the origin \((0,0)\).

Identify the transformation inside the function \(h(x) = -(x - 2)^{2}\). The expression \((x - 2)\) indicates a horizontal shift of the graph of \(f(x)\) to the right by 2 units.

Next, observe the negative sign in front of the squared term. This reflects the parabola across the x-axis, changing it from opening upwards to opening downwards.

Combine these transformations: start with the graph of \(f(x) = x^{2}\), shift it right by 2 units to get \(g(x) = (x - 2)^{2}\), then reflect it over the x-axis to get \(h(x) = -(x - 2)^{2}\).

Finally, plot the vertex of \(h(x)\) at \((2, 0)\) and sketch the parabola opening downward, maintaining the same shape as the original \(f(x)\) but transformed as described.

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Standard Quadratic Function

The standard quadratic function is f(x) = x², which graphs as a parabola opening upwards with its vertex at the origin (0,0). It serves as the base graph for understanding transformations applied to quadratic functions.

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Converting Standard Form to Vertex Form

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the base graph. For h(x) = -(x − 2)², the graph is shifted right by 2 units and reflected over the x-axis due to the negative sign.

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Intro to Transformations

Vertex Form of a Quadratic Function

The vertex form is f(x) = a(x - h)² + k, where (h, k) is the vertex. It helps identify shifts and reflections easily. In h(x) = -(x − 2)², the vertex is at (2, 0), indicating a horizontal shift and reflection.

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