# 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),

Polynomial graphs
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.

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.