Answer
please see "step by step"
Work Step by Step
Rewrite f(x) in standard form, $f(x)=a(x-h)^{2}+k$,
read the vertex, (h,x)
For the y-intercept, calcucate f(0)
For the x- intercept, solve f(x) = 0 for x.
If $a>0$, parabola opens up, the vertex is a minimum point,
If $a<0$, parabola opens down, the vertex is a maximum.
With this information (and possible additional points) sketch a graph
Read the graph for range and domain.
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a.
$f(x)=x^{2}-2x+3 $
$ f(x)=(x^{2}-2x ) +3 \quad$... complete the square
$f(x)=(x^{2}-2x+1-1)+3$
$f(x)=(x-1)^{2}+2$
b.
vertex: $(h,k)=(1, 2)$,
a=1, opens up, the vertex is a minimum
y-intercept: f(0) = $3$
x-intercepts: f(x)=0
$(x-1)^{2}+2=0$
$(x-1)^{2}=-2$
(no x-intercepts, because a square can not be negative)
c.
see image
(two pairs of additional points, either side of the vertex).
d.
domain: all reals, $\mathbb{R}$
range: $[2,\infty)$
