Describe the graph of the function f(x) = x3 − 11x2 + 36x − 36. Include the y-intercept, x-intercepts, and the shape of the graph. (30 points) will name BRAINLIEST if someone helps!!!!!
Answers
Answer:
x-intercept: (2,0),(3,0),(6,0)
y-intercept: (0,-36)
Step-by-step explanation:
Given Equation is f(x) = x³ - 11x² + 36x - 36
(i)
To find the x-intercept,substitute in 0 for y and solve for x.
y = x³ - 11x² + 36x - 36
0 = x³ - 11x² + 36x - 36
x³ - 9x² - 2x² + 18x + 18x - 36 = 0
x³ - 9x² + 18x - 2x² + 18x - 36 = 0
x(x² - 9x + 18) - 2(x²- 9x + 18) = 0
(x - 2)(x² - 9x + 18) = 0
(x - 2)(x² - 6x - 3x + 18) = 0
(x - 2)(x(x - 6) - 3(x - 6)) = 0
(x - 2)(x - 3)(x - 6) = 0
x = 2,3,6.
(ii)
To find the y-intercept, substitute in 0 for x and solve for y.
y = 0 + 11(0) + 36(0) - 36
y = 0 + 0 + 0 - 36
y = -36.
Therefore:
x-intercept : (2,0), (3,0), (6,0)
y-intercept: (0,-36)
Step-by-step explanation:
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)