Graph each polynomial function. Factor first if the expression is not in factored form. f(x)=(4x+3)(x+2)^{2}

Graph each polynomial function. Factor first if the expression is not in factored form. f(x)=(4x+3)(x+2)^{2}

Question
Polynomial graphs
asked 2021-01-10
Graph each polynomial function. Factor first if the expression is not in factored form. \(\displaystyle{f{{\left({x}\right)}}}={\left({4}{x}+{3}\right)}{\left({x}+{2}\right)}^{{{2}}}\)

Answers (1)

2021-01-11
Step 1 \(\displaystyle{f{{\left({x}\right)}}}={\left({4}{x}+{3}\right)}{\left({x}+{2}\right)}^{{{2}}}\) The function has two zeros \(\displaystyle-{\frac{{{3}}}{{{4}}}},-{2}\) So, the graph of f(x) crosses the x-axis at \(\displaystyle{\left({\frac{{{4}}}{{{3}}}},{0}\right)}\) and \(\displaystyle{\left(-{2},{0}\right)}\) To find the y-intercept, substitute 0 for x in f(x) \(\displaystyle{f{{\left({x}\right)}}}={\left({4}{x}+{3}\right)}{\left({x}+{2}\right)}^{{{2}}}\)
\(\displaystyle{f{{\left({0}\right)}}}={\left({4}{\left({0}\right)}+{3}\right)}{\left({0}+{2}\right)}^{{{2}}}\) Substitute 0 for x Simplify \(\displaystyle={\left({3}\right)}{\left({2}\right)}^{{{2}}}\)
\(\displaystyle={12}\) So, the function f(x) crosses the y-axis at (0,12) Step 2 \(\displaystyle{4}{x}\cdot{\left({x}\right)}^{{{2}}}={4}{x}^{{{3}}}\) The leading coefficient is 4 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
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