See below for the properties of the functions
How to analyze the functions?
The function f(x)
We have:
f(x) = –7(x + 2)^3(x – 1)^2(x – 3)
From the graph, we have the following features:
- Leading coefficient: Negative, because -7 is negative
- Zeros: x = -2 at multiplicity = 3, x = 1 at multiplicity = 2 and x = 3 at multiplicity = 1
- Crosses = 2
- Touches = 3
- End behavior = [tex]\mathrm{as}\:x\to \:+\infty \:,\:f\left(x\right)\to \:-\infty \:,\:\:\mathrm{and\:as}\:x\to \:-\infty \:,\:f\left(x\right)\to \:-\infty \:[/tex]
The function m(x)
We have:
m(x) = x^3 - 4x^2
Factor out x^2
m(x) = x^2(x - 4)
Factor out x
m(x) = x(x)(x - 4)
From the graph, we have the following features:
- Leading coefficient: Positive
- Zeros: x = 0 at multiplicity = 2, x = 4 at multiplicity = 1
- Crosses = 1
- Touches = 1
- End behavior = [tex]\mathrm{as}\:x\to \:+\infty \:,\:m\left(x\right)\to \:+\infty \:,\:\:\mathrm{and\:as}\:x\to \:-\infty \:,\:m\left(x\right)\to \:-\infty \:[/tex]
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