Question

2021-08-04

Step 1

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

The function has two zeros \(\displaystyle{\frac{{{1}}}{{{3}}}}\) and -2

So, the graph of f(x) crosses the x-axis at \(\displaystyle{\left({\frac{{{1}}}{{{3}}}},{0}\right)}{\quad\text{and}\quad}{\left(-{2},{0}\right)}\)

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

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

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

\(\displaystyle={\left(-{1}\right)}{\left({2}\right)}^{{{2}}}\) Simplify

\(\displaystyle=-{4}\)

So, the function f(x) crosses the y-axis at \(\displaystyle{\left({0},-{4}\right)}\)

Step 2

\(\displaystyle{3}{x}\cdot{\left({x}\right)}^{{{2}}}={3}{x}^{{{3}}}\)

The leading coefficient is 3

Since the leading coefficient is positive and the function f(x) of degree 3 (odd 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

asked 2021-08-16

Graph each polynomial function. Factor first if the expression is not in factored form. \(\displaystyle{f{{\left({x}\right)}}}={2}{x}{\left({x}-{3}\right)}{\left({x}+{2}\right)}\)

asked 2020-12-03

Rational functions can have any polynomial in the numerator and denominator. Analyse the key features of each function and sketch its graph. Describe the common features of the graphs.

\(\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ \)

\({b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ \)

\({c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}\)

asked 2021-05-02

Graph each polynomial function.

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

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

asked 2021-08-11

Graph each polynomial function. \(\displaystyle{f{{\left({x}\right)}}}={x}^{{{3}}}+{3}{x}^{{{2}}}-{4}{x}-{2}\)