Problem 64 - College Algebra 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

After applying all the transformations to the graph of standard quadratic function \(f(x)=x^2\) as per \(h(x)=\frac{1}{2}(x-1)^{2}-1\), the graph of \(h(x)\) ends up a parabola that opens upwards, moved 1 unit to the right, 1 unit down and narrower by a factor of \(\frac{1}{2}\).

Step 1: Graph the Standard Function \(f(x)=x^{2}\)

Begin by drawing the graph of the function \(f(x)=x^{2}\), which is a simple parabolic curve. This parabola opens upwards and has the vertex at the point (0,0).

Step 2: Apply the Horizontal Shift

Notice the equation \(h(x)=\frac{1}{2}(x-1)^{2}-1 \). The term (x - 1) inside the square suggests that there is a shift 1 unit to the right. So, the graph should be moved 1 unit to the right.

Step 3: Apply the Vertical Shift

The term -1 at the end of the equation suggests that there is a downward shift of 1 unit. So, the next step is to move the previously drawn graph down by 1 unit.

Step 4: Apply the Vertical Compression

The term \(\frac{1}{2} \) that's multiplied by the square suggests that there is a vertical compression by a factor of \(\frac{1}{2}\). This makes the parabola narrower. Accordingly, compress your graph vertically by a factor of \(\frac{1}{2} \).