Polynomial functions: f(x)=(x-1) (x-2)^2-- NEED HELP PLEASE!!? Find end behavior

Question

Polynomial functions: f(x)=(x-1) (x-2)^2-- NEED HELP PLEASE!!?

Find the end behavior and explain how it was done

Find the x-intercept(s) and show how it was done

Find the y-intercept(s) and show how it was done.

List intervals determined by the x-intercepts

Select at least one x-value in each interval and show how it indicates if the graph is positive or negative within that interval.

Please explain how to do this or give answers in details for best anwer!! I know it is a lot but I really need help and would really appreciate it thanks!!

Answer 1

For f(x)=(x-1)(x-2)²: (red graph)

End behaviour: f(x)→x³ when x is very large. Simply ignore the constants 1 and 2, so we just add the powers of x. That gives us 3. When x is large and positive, f(x)→∞; when x is large and negative, f(x)→−∞.

x-intercepts: x=1 and x=2 because these values make f(x)=0.

y-intercepts: (-1)(-2)²=-4. There is only one y-intercept at y=-4. This is the value of f(0).

(a) Intervals: (−∞,1), f(x)<0 (below the x-axis); (b) [1,2], f(x)≥0 (on or above the x-axis); (c) (2,∞), f(x)>0 (above the x-axis).

(a) example: f(-1)=-18; (b) example: f(1.5)=0.125; (c) example: f(3)=2.

For f(x)=-2(x-3)(x+5)(x-2)²: (blue graph)

End behaviour: f(x)=-2x⁴. When x→±∞, f(x)→−∞.

x-intercepts: -5, 2, 3

y-intercept: -2(-3)(5)(4)=120

Intervals: 

(−∞,-5): f(x) is (-)(-)(-)(+)=negative; example: f(-6)=(-2)(-9)(-1)(64)=-1152.

[-5,2]: f(x) is (-)(-)(+)(+)≥0; example: f(1)=(-2)(-2)(6)(1)=24.

[2,3]: f(x) is (-)(-)(+)(+)≥0; example: f(2.5)=(-2)(-0.5)(7.5)(0.25)=1.875.

(3,∞): f(x) is (-)(+)(+)(+)≤0; example: f(4)=(-2)(1)(9)(4)=-72.