Question

Can you please help me with 44 Also please use all 3 forms of the expression : up/down, as _ ,_ and limits

Answer 1

Solving #44.

The expression we have is:

`h(x)=(x-1)^3(x+3)^2`

Graph of the function:

Finding the x-intercepts.

The x-intercepts can be found by making the expression equal to 0:

`(x-1)^3(x+3)^2=0`

Then we make each parenthesis equal to 0:

`\begin{gathered} x-1=0 \\ x+3=0 \end{gathered}`

Solving for x:

`\begin{gathered} x=1 \\ x=-3 \end{gathered}`

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:

`(x-1)^3(x+3)^2\longrightarrow(0-1)^3(0+3)^2=(-1)^3(3)^2=(-1)(9)=-9`

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 __,__

`\begin{gathered} as\text{ x}\longrightarrow\infty,h(x)\longrightarrow\infty \\ as\text{ x}\longrightarrow-\infty,h(x)\longrightarrow-\infty \end{gathered}`

Form 3. Limits

`\begin{gathered} \lim _(x\to\infty)h(x)=\infty \\ \lim _(x\to-\infty)h(x)=-\infty \end{gathered}`

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 __,__

`\begin{gathered} as\text{ x}\longrightarrow\infty,h(x)\longrightarrow\infty \\ as\text{ x}\longrightarrow-\infty,h(x)\longrightarrow-\infty \end{gathered}`

Form 3. Limits

`\begin{gathered} \lim _(x\to\infty)h(x)=\infty \\ \lim _(x\to-\infty)h(x)=-\infty \end{gathered}`