Question

Graph each polynomial function. Factor first if the expression is

Polynomial graphs
ANSWERED
asked 2021-08-16
Graph each polynomial function. Factor first if the expression is not in factored form. \(\displaystyle{f{{\left({x}\right)}}}={2}{x}{\left({x}-{3}\right)}{\left({x}+{2}\right)}\)

Answers (1)

2021-08-17
Step 1
\(\displaystyle{f{{\left({x}\right)}}}={2}{x}{\left({x}-{3}\right)}{\left({x}+{2}\right)}\) The function in factored form
The function has three zeros 0,3, and -2
So, the graph of f(x) crosses the x-axis at (0,0),(3,0), and (-2,0)
To find the y-intercept, substitute 0 for x in f(x)
\(\displaystyle{f{{\left({x}\right)}}}={2}{x}{\left({x}-{3}\right)}{\left({x}+{2}\right)}\)
\(\displaystyle{f{{\left({0}\right)}}}={2}{\left({0}\right)}{\left({0}-{3}\right)}{\left({0}+{2}\right)}\) Substitute 0 for x
\(\displaystyle={0}\)
So, the function f(x) crosses the y-axis at (0,0)
Step 2
\(\displaystyle{2}{x}\cdot{x}\cdot{x}={2}{x}^{{{3}}}\)
The leading coefficient is 2
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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