Problem 62 - 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. $$g(x)=\frac{1}{2}(x-1)^{2}$$

Step-by-Step Solution

Answer

The graph of \(g(x)=\frac{1}{2}(x-1)^{2}\) is a parabola that is wider than the graph of \(f(x)=x^{2}\) and is shifted one unit to the right.

Step 1: Graph the Standard Quadratic Function

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

Step 2: Apply Horizontal Shift

Now apply the transformation \(x \rightarrow x-1\). This means shifting every point on the curve of \(f(x)=x^{2}\) one unit to the right.

Step 3: Apply Vertical Stretch

Next, apply the transformation \(f(x) \rightarrow \frac{1}{2} f(x)\). This involves stretching the graph vertically by a factor of 1/2. The negative parabola now becomes wider.

Step 4: Graph the Transformed Function

Plot the transformed points to get the graph of \(g(x)=\frac{1}{2}(x-1)^{2}\). The parabola \(g(x)=\frac{1}{2}(x-1)^{2}\) is wider than the original one and its vertex is shifted one unit to the right.