Graph each polynomial function. Factor first if the expression is not in factored form. f(x)=(3x-1)(x+2)^{2}

Graph each polynomial function. Factor first if the expression is not in factored form. f(x)=(3x-1)(x+2)^{2}

Question
Polynomial graphs
asked 2021-01-13
Graph each polynomial function. Factor first if the expression is not in factored form.
\(\displaystyle{f{{\left({x}\right)}}}={\left({3}{x}-{1}\right)}{\left({x}+{2}\right)}^{{{2}}}\)

Answers (1)

2021-01-14
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 -2ZSK
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
image
0

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