Find the vertex of the qudratic function f(x)=x^2-6x+42, then express the qudratic function in standart form f(x)=a(x-h)^2+k and state whether the vertex is a minimum or maximum. Enter exat answers only, no approximations.

kuCAu 2021-03-01 Answered
Find the vertex of the qudratic function \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{6}{x}+{42}\), then express the qudratic function in standart form \(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}-{h}\right)}^{{2}}+{k}\) and state whether the vertex is a minimum or maximum. Enter exat answers only, no approximations.

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Expert Answer

curwyrm
Answered 2021-03-02 Author has 15665 answers

To express the given quadratic function in standard form \(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}−{h}\right)}^{{2}}+{k}\)
Re-write the given quadratic function as:
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}−{6}{x}+{42}\)
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}−{6}{x}+{9}+{33}\)
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}−{2}×{3}{x}+{3}^{{2}}+{33}\)
\(\displaystyle{f{{\left({x}\right)}}}={\left({x}−{3}\right)}^{{2}}+{33}\)
The standard form of a given quadratic function is \(\displaystyle{f{{\left({x}\right)}}}={\left({x}−{3}\right)}^{{2}}+{33}\)
Here
\(a=1, h=3,\) and \(k=33.\)
The given quadratic function is an upward parabola with vertex (3, 33) and it has a minimum at the vertex.
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