Question

Describe the graph of the function f(x) = x^3 - 18x^2 + 101x - 180 Include y- intercept, x- intercept, and the shape of the graph

Answer 1

Let's first calculate the roots of f(x) = x³-18x²+101x-180
a) f(4) = 4³ - 18.(4²) +101.(4) -180 =0, then x=4 is a root.
b) to find the 2 other roots divide x³-18x²+101x-180 by (x-4) = x²-14x+45
c) this quadratic equation has the following roots: x=5 and x=9. Hence:
 
f(x) = x³-18x²+101x-180 = (x-4)(x-5)(x-9)
the x-intercepts are x₁ = 4,   x₂ = 5,   x₃ = 9.
To find the y intercept plug x= 0 in f(x) =  x³-18x²+101x-180

f(x) = y = -180 (y intercept

The graph starts from - ∞ to + ∞ and passes through one maximum and one minimum