Analyze the polynomial function f(x)= -3(x+2)(x-2)^3 using parts (a) through (e).
a) Determine the end behavior of the graph of the function.
The graph of f behaves like y= for large values of x
Since the leading term of the polynomial function is -3(x^4), the end behavior of the graph of the function resembles that of a fourth degree polynomial: as x approaches positive or negative infinity, the graph will rise to positive infinity.
b) State the leading term of the function.
The leading term of the function is -3x^4.
c) Determine the degree of the function.
The degree of the function is 4.
d) Find the y-intercept of the function.
To find the y-intercept, we plug in x=0 into the function:
f(0) = -3(0+2)(0-2)^3
f(0) = -3(2)(-8)
f(0) = 48
Therefore, the y-intercept is (0, 48).
e) Determine the x-intercepts of the function.
To find the x-intercepts, we set f(x) = 0 and solve for x:
-3(x+2)(x-2)^3 = 0
x+2 = 0 or (x-2)^3 = 0
x = -2 or x = 2
Therefore, the x-intercepts are (-2, 0) and (2, 0).