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. Thanks!

Answer 1

For a polyonomial
P(x)=ax^n+bx^(n-1)...zx^0
when n is odd, the endpoints of the graph point in oposite directions
when n is even, the endpoints of the graph point in same direction

a is the leading term
when a is positive and:
1. n is odd, the graph goes from bottom left to top right
2. n is even, the graph goes from top left to top right

when a is negative and:
1. n is odd, the graph goes from top left to bottom right
2. n is even, the graph goes from bottom left to bottom right



xintercept is where the line crosses the x axis or when y=0
yintercept is where the line crosses the y axis or when x=0




f(x)=1x^3-11x^2+36x-36
leading term is positive and odd degree

graph goes from bottom left to top right, has 1 inflection point

yint, sub 0 for all x's
f(0)=0^3-11(0)^2+36(0)-36
yintercept is y=-36 or (0,-36)

xintercept
set function equal to zero
0=x^3-11x^2+36x-36
factor
0=(x-2)(x-3)(x-6)
x=2,3,6
xintercepts are at x=2,3,6 or (2,0), (3,0) and (6,0)

Answer 2

X-intercepts: 2,   3,    6.
Y-intercepts: -36.
The graph is included as an image.