Problem 64 - Precalculus Essentials

Question

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$h(x)=\frac{1}{2}(x-1)^{2}-1$$

Step-by-Step Solution

Answer

The graph of the given function is a parabola that is narrower than the graph of the standard quadratic function, shifted one unit to the right and one unit down.

Step 1: Graph the Standard Quadratic Function

Begin by graphing the standard quadratic function which is \(f(x) = x^2\). It is a parabola that opens upwards with its vertex at the origin, (0,0).

Step 2: Identify the Transformations

In the function \(h(x) = \frac{1}{2}(x-1)^2 - 1\), the (x-1) shifts the graph one unit to the right and -1 shifts the graph one unit down. The \(\frac{1}{2}\) in front of the parenthesis is a vertical shrink by a factor of \(\frac{1}{2}\). It causes the graph of \(h(x)\) to be narrower than that of \(f(x)\).

Step 3: Apply the Transformations

Apply the transformations as identified in the previous step. This involves shifting the graph of \(f(x)\) one unit to the right, one unit down and make it narrower with a factor of \(\frac{1}{2}\).

Step 4: Graph the Transformed Function

Draw the transformed function \(h(x) = \frac{1}{2}(x-1)^2 - 1\), making sure to show the transformations from the standard quadratic function.