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 by -200,200
b. -50,50 by -10,10
c. -10,10 by -10,10
d. -100,100 by -100,100
lim f(x) as x→∞

Explanation:

Step1: Find x - intercepts

Set \(f(x)=(x - 3)^{2}(x + 4)(x - 1)=0\). Then \(x=3\) (with multiplicity 2), \(x=-4\) and \(x = 1\). The x - intercepts are \(x=-4,1,3\). We need a viewing window that includes these values.

Step2: Consider end - behavior

The function \(f(x)=(x - 3)^{2}(x + 4)(x - 1)\) is a polynomial of degree \(2 + 1+1=4\) (since \((x - 3)^{2}\) has degree 2, \((x + 4)\) has degree 1 and \((x - 1)\) has degree 1) and the leading coefficient is positive (the product of the leading coefficients of each factor: \(1\times1\times1 = 1\)). So \(\lim_{x
ightarrow\pm\infty}f(x)=+\infty\).

Step3: Analyze viewing windows

Option A: \([-5,5]\) by \([-200,200]\) includes the x - intercepts \(x=-4,1,3\) and can show the extrema and the end - behavior well. Option B: \([-50,50]\) by \([-10,10]\) may not show the end - behavior and the function values well as the y - range is too small. Option C: \([-10,10]\) by \([-10,10]\) has a small y - range. Option D: \([-100,100]\) by \([-100,100]\) is too large and may not show the details of the extrema and x - intercepts clearly.

Step4: Find \(\lim_{x

ightarrow\infty}f(x)\)
Since \(f(x)\) is a polynomial of degree 4 with a positive leading coefficient, \(\lim_{x
ightarrow\infty}f(x)=+\infty\).

Answer:

A. \([-5,5]\) by \([-200,200]\)
\(\lim_{x
ightarrow\infty}f(x)=+\infty\)