Question

graph the function in a viewing - window that shows all of its extrema and x - intercepts. describe the end behavior using limits.
f(x)=(x - 3)^{2}(x + 4)(x - 1)
which of the following viewing windows gives the following comprehensive graph?
a. (-5,5\text{ by }-200,200) b. (-50,50\text{ by }-10,10) c. (-10,10\text{ by }-10,10) d. (-100,100\text{ by }-100,100)
(lim_{x
ightarrowinfty}f(x)=infty)
(lim_{x
ightarrow-infty}f(x)=square)

Explanation:

Step1: Determine the degree of the polynomial

The function $f(x)=(x - 3)^2(x + 4)(x - 1)$ is a polynomial. Expand it: $(x - 3)^2=x^{2}-6x + 9$, then $f(x)=(x^{2}-6x + 9)(x + 4)(x - 1)$. Multiply $(x^{2}-6x + 9)(x + 4)=x^{3}+4x^{2}-6x^{2}-24x+9x + 36=x^{3}-2x^{2}-15x + 36$, and then multiply by $(x - 1)$ to get a fourth - degree polynomial. The leading coefficient is positive (since the product of the leading coefficients of each factor is positive: $1\times1\times1 = 1$).

Step2: Analyze the x - intercepts

Set $f(x)=0$. Then $(x - 3)^2(x + 4)(x - 1)=0$. The x - intercepts are $x = 3$ (with multiplicity 2), $x=-4$ and $x = 1$.

Step3: Analyze the viewing window

We need a window that includes the x - intercepts $x=-4,1,3$. Option A: $[-5,5]$ by $[-200,200]$ includes these x - values and is likely to show the extrema well. Options B, C and D have either too large or too small intervals for the x - values or y - values that may not show the extrema and x - intercepts comprehensively.

Step4: Analyze the end - behavior

For a fourth - degree polynomial with a positive leading coefficient, $\lim_{x
ightarrow\infty}f(x)=\infty$ and $\lim_{x
ightarrow-\infty}f(x)=\infty$.

Answer:

A. $[-5,5]$ by $[-200,200]$, $\infty$