$\newline$ Given the function $f(x)=\frac{(x+4)^{2}(x-2)}{2(x-1)^{2}(x-3)}$ , use the characteristics of polynomials and rational functions to describe its behavior and sketch the function. $\newline$ Enter the exact answers. $\newline$ Enter the intercepts as points, $(a, b)$ . Enter the $x$ -intercepts in increasing order of the $x$ -coordinate. $\newline$ The $x$ -intercepts are $\square$ and $\square$ $\newline$ The $y$ -intercept is $\square$ $\newline$ The fields below accept a list of numbers or formulas separated by semicolons (e.g. $2 ; 4 ; 6$ or $(a, b)$ $0$ . The order of the list does not matter.

Full solution

\newline

Given the function

f(x)=\frac{(x+4)^{2}(x-2)}{2(x-1)^{2}(x-3)}

, use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.

\newline

Enter the exact answers.

\newline

Enter the intercepts as points,

(a, b)

. Enter the

x

-intercepts in increasing order of the

x

-coordinate.

\newline

The

x

-intercepts are

\square

and

\square

\newline

The

y

-intercept is

\square

\newline

The fields below accept a list of numbers or formulas separated by semicolons (e.g.

2 ; 4 ; 6

(a, b)

0

. The order of the list does not matter.

Identify x-intercepts: Identify the x-intercepts by setting the numerator equal to zero and solving for $x$ :
$(x + 4)^2 * (x - 2) = 0$
Solve $(x + 4)^2 = 0$ $\rightarrow x = -4$ (double root)
Solve $(x - 2) = 0$ $\rightarrow x = 2$
Identify y-intercept: Identify the y-intercept by setting $x = 0$ in the function: $\newline$ f(0) = \frac{(0 + 4)^2 * (0 - 2)}{2 * (0 - 1)^2 * (0 - 3)}$\newline = \frac{(16 * -2)}{(2 * 1 * -3)} $\newline$ = \frac{-32}{-6} $\newline$ = \frac{32}{6} $\newline$ = \frac{16}{3}$
Identify vertical asymptotes: Identify vertical asymptotes by setting the denominator equal to zero and solving for $x$ : $2 \cdot (x - 1)^2 \cdot (x - 3) = 0$ Solve $(x - 1)^2 = 0 \rightarrow x = 1$ (double root)Solve $(x - 3) = 0 \rightarrow x = 3$
Identify horizontal asymptotes: Identify horizontal asymptotes by comparing the degrees of the numerator and the denominator: $\newline$ Degree of numerator = $3$ (from $(x + 4)^2 * (x - 2)$ ) $\newline$ Degree of denominator = $3$ (from $2 * (x - 1)^2 * (x - 3)$ ) $\newline$ Since the degrees are equal, the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{1}{2}$ .