Solving #44.
The expression we have is:
[tex]h(x)=(x-1)^3(x+3)^2[/tex]
Graph of the function:
Finding the x-intercepts.
The x-intercepts can be found by making the expression equal to 0:
[tex](x-1)^3(x+3)^2=0[/tex]
Then we make each parenthesis equal to 0:
[tex]\begin{gathered} x-1=0 \\ x+3=0 \end{gathered}[/tex]
Solving for x:
[tex]\begin{gathered} x=1 \\ x=-3 \end{gathered}[/tex]
The multiplicity of each one if given by the exponent that their parentheses had:
x-intercepts: 1 (multiplicity of 3) and -3 (multiplicity of 2).
Finding the y-intercept.
To find the y-intercept we make the x equal to 0 in the expression:
[tex](x-1)^3(x+3)^2\longrightarrow(0-1)^3(0+3)^2=(-1)^3(3)^2=(-1)(9)=-9[/tex]
y-intercept: -9
Finding the end-behavior.
Again we consider the graph of the function:
Form 1. Ups/downs
As x increases the function goes up and as x decreases the function goes down
Form 2. as __,__
[tex]\begin{gathered} as\text{ x}\longrightarrow\infty,h(x)\longrightarrow\infty \\ as\text{ x}\longrightarrow-\infty,h(x)\longrightarrow-\infty \end{gathered}[/tex]
Form 3. Limits
[tex]\begin{gathered} \lim _{x\to\infty}h(x)=\infty \\ \lim _{x\to-\infty}h(x)=-\infty \end{gathered}[/tex]
Answer:
x-intercepts: 1 (multiplicity of 3) and -3 (multiplicity of 2).
y-intercept: -9
End-behavior:
Form 1. Ups/downs
As x increases the function goes up and as x decreases the function goes down
Form 2. as __,__
[tex]\begin{gathered} as\text{ x}\longrightarrow\infty,h(x)\longrightarrow\infty \\ as\text{ x}\longrightarrow-\infty,h(x)\longrightarrow-\infty \end{gathered}[/tex]
Form 3. Limits
[tex]\begin{gathered} \lim _{x\to\infty}h(x)=\infty \\ \lim _{x\to-\infty}h(x)=-\infty \end{gathered}[/tex]
