Chapter 1, Problem 13RE

To determine

To sketch: The graph of the function $y=x^2-2x+2$ by using transformation.

Explanation of Solution

Given:

The function is, $y=x^2-2x+2$.

Graph:

Express the given function as follows.

$\begin{array}{c}y=x^2-2x+2\\ =x^2-2x+1+1\\ ={\left(x-1\right)}^2+1\end{array}$

Thus, draw the graph of $y={\left(x-1\right)}^2+1$.

The standard graph of the function $y=x^2$ is roughly drawn as shown below in Figure 1.

Then, draw the graph of $y={\left(x-1\right)}^2$, shift the graph $y=x^2$ by 1 unit to the right. Thus, the graph of $y={\left(x-1\right)}^2$ is shown below in Figure 2.

From Figure 2, it is observed that Figure 1 is shifted 1 unit to the right.

To draw the graph of $y={\left(x-1\right)}^2+1$, shift the graph $y={\left(x-1\right)}^2$ by 1 unit upward. Thus, the graph of $y={\left(x-1\right)}^2+1$ is shown below in Figure 3.

Observe that Figure 3 is obtained from Figure 1 in such a way that it is shifted 1 unit to the right and then shifted 1 unit upward.