Step 1

Given:

\(\displaystyle{y}={1}-{2}{x}+{x}^{{2}}\)

To sketch:

The graph of \(\displaystyle{y}={1}-{2}{x}+{x}^{{2}}\) using transformations.

Step 2

First we need to identify the parent function of \(\displaystyle{y}={1}-{2}{x}+{x}^{{2}}\)

\(\displaystyle{y}={1}-{2}{x}+{x}^{{2}}\)

\(\displaystyle{y}={x}^{{2}}-{2}{x}+{1}\)

\(\displaystyle{y}={\left({x}-{1}\right)}^{{2}}\)

Here the parent function is \(\displaystyle{y}={x}^{{2}}\) . Using this, we will get the translation graph \(\displaystyle{y}={\left({x}-{1}\right)}^{{2}}\)

Since, \(\displaystyle{y}={\left({x}-{1}\right)}^{{2}}\) is a parabola with vertex at (1,0) .

Step 3

The graph of \(\displaystyle{y}={x}^{{2}}\) of vertex (0,0) is given by,

The graph of \(y=(x-1)^2\) is obtained by shifting the graph \(\displaystyle{y}={x}^{{2}}\) by 1 unit in the right direction.

Thus, the graph of \(\displaystyle{y}={1}-{2}{x}+{x}^{{2}}\) is given by,