Question

Describe the graph of the function f(x) = x3 − 11x2 + 36x − 36. Include the y-intercept, x-intercepts, and the shape of the graph.

Answer 1

The given function is
f(x) = x³ - 11x² + 36x - 36
When x = 0, f = -36
The y-intercept is -36

Possible zeros are +/-1, +/-2, +/-3, +/-4, +/-6, +/-9
From the Remainder Theorem, x=a is a zerp if f(a) = 0.
Try x = 1:
f(1) = 1 - 11 + 36 - 36 = -10 (not a zero)
Try x = 2:
f(2) = 8 - 44 + 72 - 36  (this is a zero).

Use synthetic division.
2|  1  -11    36  -36
           2  -18    36
   ----------------------
     1   -9   18      0

Therefore
f(x) = (x - 2)(x² - 9x + 18)
      = (x - 2)(x - 3)(x - 6)
The x-intercepts are x = 2, 3, 6

As x→ -∞, f→ -8
As x→ +∞, f→ +∞

To find how f(x) behaves between the zeros, test function values between zeros.

   x      f(x)    Comments
----   ---------   ---------------
  -1       -84      negative
   0      -36      negative
   2         0       zero
 2.5  0.875      positive
   3          0      zero
   4         -4      negative
   6          0      zero
   8         60     positive
   
Summary:
y-intercept = -36
x-intercepts: 2.3.6
f→-∞ as x→-∞
f→+∞ as x→+∞

We now know enough about the shape of the curve to sketch it, as shown below.