Question

Given the equation below graph the polynomial.Indicate x and y intercepts, multiplicity and the end behaviors

Answer 1

Given:

$h(x)=(x+3)^2(x-2)$

You can rewrite it as follows:

$y=(x+3)^2(x-2)$

You can graph it by following these steps:

1. Find the x-intercepts. By definition, the value of "y" is zero when the function intersects the x-axis. Then, you need to make:

$y=0$

Substitute this value into the equation:

$0=(x+3)^2(x-2)$

Solving for "x", you get:

$\begin{cases}(x+3)^2=0\Rightarrow x+3=0\Rightarrow x_1=-3 \\ \\ (x-2)=0\Rightarrow x_2=2\end{cases}$

2. Identify the multiplicities:

- Notice that the first factor of the function is:

$(x+3)^2$

And has an exponent 2.

This means that:

$\text{ }x_1=-3\rightarrow Multiplicity\text{ }2$

Since it has an even Multiplicity, the graph touches the x-axis, at that point, but it does not intersect it.

- Having the other factor:

$(x-2)$

Its exponent is 1. Then:

$\text{ }x_2=2\rightarrow Multiplicity\text{ }1$

That even Multiplicity indicates that the graph intersects the x-axis at that point.

3. Find the y-intercept. By definition, the value of "x" is zero when the function intersects the y-axis. Then, you need to make:

$x=0$

Substitute this value into the function and solve for "y":

$\begin{gathered} y=(0+3)^2(0-2) \\ y=(9)(-2) \\ y=-18 \end{gathered}$

4. By definition, the end behavior of a function can be determined knowing its Leading Coefficient and its degree. Then, you need to follow these steps to find the end behaviors of the function:

- Expand the right side of the equation. Remember this formula:

$(a+b)^2=a^2+2ab+b^2$

Then, you get:

$\begin{gathered} h(x)=(x^2+2(x)(3)+3^2)(x-2) \\ \\ h(x)=(x^2+6x+9)(x-2) \end{gathered}$

Multiply the polynomials by applying the Distributive Property:

$h(x)=(x^2)(x)+(6x)(x)+9(x)-2(x^2)-(2)(6x)-(2)(9)$$h(x)=x^3+6x^2+9x-2x^2-12x-18$

Add the like terms:

$h(x)=x^3+4x^2-3x-18$

Knowing that the degree is the highest exponent and that the Leading Coefficient "a" is the number that multiplies the term with the highest exponent, you can identify that:

$\begin{gathered} a=1 \\ Degree=3 \end{gathered}$

By definition, if:

$a>0$

The end behaviors of a function are:

$As\text{ }x\to+\infty,\text{ }f\mleft(x\mright)\to\infty$$As\text{ }x\to-\infty,\text{ }f(x)\to-\infty$

Therefore, in this case, since the Leading Coefficient is positive, you can determine that as "x" approaches positive infinity, h(x) approaches positive infinity:

$As\text{ }x\to+\infty,\text{ }h(x)\to+\infty$

And as "x" approaches negative infinity, h(x) approaches negative infinity:

$As\text{ }x\to-\infty,\text{ }h(x)\to-\infty$

Knowing all the data found, you can graph the Cubic Function.

Hence, the answers are:

• x-intercepts:

$\begin{gathered} x_1=-3 \\ x_2=2 \end{gathered}$

• y-intercept:

$y=-18$

• Multiplicity:

$\begin{gathered} \text{ }x_1=-3\rightarrow Multiplicity\text{ }2 \\ \\ \text{ }x_2=2\rightarrow Multiplicity\text{ }1 \end{gathered}$

• End behaviors:

$\begin{gathered} As\text{ }x\to+\infty,\text{ }h(x)\to+\infty \\ \\ As\text{ }x\to-\infty,\text{ }h(x)\to-\infty \end{gathered}$

• Graph: