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For the following exercise, for each polynomial f(x)= frac{1}{2}x^{2} - 1: a) find the degree, b) find the zeros, if any, c) find the y-intercept(s),

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asked 2020-11-01
For the following exercise, for each polynomial \(\displaystyle{f{{\left({x}\right)}}}=\ {\frac{{{1}}}{{{2}}}}{x}^{{{2}}}\ -\ {1}\): a) find the degree, b) find the zeros, if any, c) find the y-intercept(s), if any, d) use the leading coefficient to determine the graph’s end behavior, e) determine algebraically whether the polynomial is even, odd, or neither.

Answers (1)

2020-11-02
a) Since degree of polynomial is the greatest power of x, then the degree of \(\displaystyle{f{{\left({x}\right)}}}=\ {\frac{{{1}}}{{{2}}}}{x}^{{{2}}}\ -\ {1}\ {i}{s}\ {2}\) b) To find the zeros: \(\displaystyle{f{{\left({x}\right)}}}={0}\)
\(\displaystyle{\frac{{{1}}}{{{2}}}}{x}^{{{2}}}\ -\ {1}={0}\)
\(\displaystyle{x}^{{{2}}}\ -\ {2}={0}\) Then: \(\displaystyle{x}=\ \pm\ \sqrt{{{2}}}\) c) To find y-intercept, put \(\displaystyle{x}={0}\), then y-intercept is -1. d) Here \(\displaystyle{n}={2},\ \text{even and}\ {a}_{{{n}}}=\ {\frac{{{1}}}{{{2}}}}\ {>}\ {0}\), then graph rises to the left and right. e) To determine whether the polinomial is even, odd, or neither, replace z with -x: \(\displaystyle{f{{\left(-{x}\right)}}}=\ {\frac{{{1}}}{{{2}}}}{\left(-{x}\right)}^{{{2}}}\ -\ {1}\)
\(\displaystyle={f{{\left({x}\right)}}}\) Then function is even.
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