Question

In the following exercises, graph each equation.

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

Short answer

Graph for the equation given will be a parabola.

Step-by-step solution

Identify the conic from the equation,

Equation given in the question,

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

Since, variable $x$is squared only,

Equation will represent a parabola.

Find x, y intercepts of the graph.

For $x-$intercept,

Put $y=0$in the equation.

role="math" localid="1645382935400" $$\begin{gathered} 0={(x-1)}^2+2 \\ {(x-1)}^2=-2 \\ (x-1)=\sqrt{-2} \end{gathered}$$

Since, $\sqrt{-2}$is not defined, $x-$intercepts will be undefined.

Hence, there will be no $x-$intercept.

For $y-$intercepts,

Put $x=0$in the equation.

$y={(0-1)}^2+2$

$y=3$

There will be one $y-$intercept only.

Find vertex and other points on the parabola.

Compare the given equation with the vertex form of the equation of a parabola.

$y={(x-h)}^2+k$

Where, $(h,k)$is the vertex.

Therefore, vertex of the given equation will be $(1,2)$.

Table for other points,

role="math" localid="1645383724764" $x$ $-1$ $0$ $1$ $2$

$y$ $6$ $3$ $2$ $3$

Graph the equation.