 # A quadratic function f is given (a)Express f in standard form (b)Find the vertex and x- and y-intercepts of f. (c)Sketch the graph.(d)Find the domain and range of f.f(x)=x^{2}-2x+3 preprekomW 2021-09-14 Answered
A quadratic function f is given
(a)Express f in standard form
(b)Find the vertex and x- and y-intercepts of f.
(c)Sketch the graph.
(d)Find the domain and range of f.
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}-{2}{x}+{3}$$

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$$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}−{2}{x}+{3}$$The standard form of quadratic equation is $$\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}−{h}\right)}{2}+{k}$$
(1)(a)$$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}−{2}{x}+{1}+{2}.{f{{\left({x}\right)}}}={\left({x}−{1}\right)}{2}+{2}$$
On comparing it with equation (1) we get:$$h=1$$,$$k=2(b)$$ The vertex V is given by $$V=(h,k)=(1,2)$$ The y−intercept is given by $$x=0$$

$$f(0)=3$$

So, we have (0,3) as the y−intercept.

Now the x−intercept is given by equating $$\displaystyle{f{{\left({x}\right)}}}={0}$$
$$\displaystyle{x}^{{{2}}}−{2}{x}+{3}={0}$$
$$\displaystyle{x}^{{{2}}}−{3}{x}+{x}−{3}={0}{x}$$
$$\displaystyle{\left({x}−{3}\right)}+{1}{\left({x}−{3}\right)}={0}$$
$$\displaystyle{x}=−{1},{3}{S}{o},{x}−int{e}{r}{c}{e}{p}{t}{s}\ {o}{f}\ {f}{a}{r}{e}{\left(−{1},{0}\right)}{\quad\text{and}\quad}{\left({3},{0}\right)}$$