Step 1

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}-{4}\right)}{\left({x}-{1}\right)}\)

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}-{2}\right)}{\left({x}+{2}\right)}{\left({x}-{1}\right)}\) The function in factored form

The function has four zeros 0,2,-2 and 1

So, the graph of f(x) crosses the x-axis at (0,0), (2,0), (-2,0) and (1,0)

To find the y-intercept, subtitute 0 for x in f(x)

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}-{4}\right)}{\left({x}-{1}\right)}\)

\(\displaystyle{f{{\left({0}\right)}}}={2}{\left({0}\right)}^{{{3}}}{\left({0}^{{{2}}}-{4}\right)}{\left({0}-{1}\right)}\) Substitute 0 for x

\(\displaystyle={0}\)

So, the function f(x) crosses the y-axis at (0,0

Step 2 \(\displaystyle{2}{x}^{{{3}}}\cdot{x}^{{{2}}}\cdot{x}={2}{x}^{{{6}}}\)

The leading coefficient is 2

Since the leading coefficient is positive and the function f(x) of degree 6 (even degree)

So, the end behavior is

\(\displaystyle{x}\rightarrow\infty,{f{{\left({x}\right)}}}\rightarrow\infty\)

\(\displaystyle{x}\rightarrow-\infty,{f{{\left({x}\right)}}}\rightarrow\infty\)

See the graph below

\(\displaystyle{h}{\mathtt{{p}}}{s}:{/}{q}{2}{a}.{s}{3}-{u}{s}-{w}{e}{s}{t}-{1}.{a}{m}{a}{z}{o}{n}{a}{w}{s}.{c}{o}\frac{{m}}{{d}}{e}\frac{{v}}{{19610300981}}.{j}{p}{g}\)

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}-{4}\right)}{\left({x}-{1}\right)}\)

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}-{2}\right)}{\left({x}+{2}\right)}{\left({x}-{1}\right)}\) The function in factored form

The function has four zeros 0,2,-2 and 1

So, the graph of f(x) crosses the x-axis at (0,0), (2,0), (-2,0) and (1,0)

To find the y-intercept, subtitute 0 for x in f(x)

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}-{4}\right)}{\left({x}-{1}\right)}\)

\(\displaystyle{f{{\left({0}\right)}}}={2}{\left({0}\right)}^{{{3}}}{\left({0}^{{{2}}}-{4}\right)}{\left({0}-{1}\right)}\) Substitute 0 for x

\(\displaystyle={0}\)

So, the function f(x) crosses the y-axis at (0,0

Step 2 \(\displaystyle{2}{x}^{{{3}}}\cdot{x}^{{{2}}}\cdot{x}={2}{x}^{{{6}}}\)

The leading coefficient is 2

Since the leading coefficient is positive and the function f(x) of degree 6 (even degree)

So, the end behavior is

\(\displaystyle{x}\rightarrow\infty,{f{{\left({x}\right)}}}\rightarrow\infty\)

\(\displaystyle{x}\rightarrow-\infty,{f{{\left({x}\right)}}}\rightarrow\infty\)

See the graph below

\(\displaystyle{h}{\mathtt{{p}}}{s}:{/}{q}{2}{a}.{s}{3}-{u}{s}-{w}{e}{s}{t}-{1}.{a}{m}{a}{z}{o}{n}{a}{w}{s}.{c}{o}\frac{{m}}{{d}}{e}\frac{{v}}{{19610300981}}.{j}{p}{g}\)