Problem 60 - 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. $$h(x)=(x-1)^{2}+2$$

Step-by-Step Solution

Answer

To graph \(h(x) = (x-1)^2 + 2\), first graph the standard quadratic function \(f(x) = x^2\). Then shift the graph 1 unit to the right and 2 units upwards. The resulting graph is a parabola centered at the point (1,2). This is the plot for \(h(x)\).

Step 1: Graph the Standard Quadratic Function

Begin by graphing the standard quadratic function \(f(x) = x^2\). This is a parabola centered at the origin with its arms opening upwards.

Step 2: Apply Horizontal Shift

The term \((x-1)^2\) in \(h(x)\) is a horizontal shift of the graph of \(f(x)\) to the right by 1 unit. Therefore, shift every point on the graph of \(f(x)\) one unit to the right.

Step 3: Apply Vertical Shift

The \(+2\) in \(h(x)\) is a vertical shift of the graph of \(f(x)\) upwards by 2 units. Therefore, after applying the horizontal shift, now shift every point on this new graph two units upwards.

Step 4: Draw the Transformed Graph

After applying the transformations, draw the resulting graph which represents the function \(h(x) = (x-1)^2 + 2\). The new parabola should be centered at the point (1,2) with its arms still opening upwards.