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Chapter 2: Problem 60

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.
04

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.

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