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
. The leading term in the polynomial is -3(x-2)^3, which will dominate the function as x approaches positive or negative infinity. Since the leading term has an odd degree of 3 and a negative coefficient, the end behavior of the graph will be as follows:
- As x approaches positive infinity, the leading term will dominate and the function will decrease without bound, approaching negative infinity.
- As x approaches negative infinity, the leading term will again dominate and the function will increase without bound, approaching positive infinity.
b) Find the x- and y-intercepts of the graph of the function.
To find the x-intercepts, set f(x) equal to zero 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 x = -2 and x = 2.
To find the y-intercept, set x equal to 0:
f(0) = -3(0+2)(0-2)^3
f(0) = -3(2)(-8)
f(0) = 48
Therefore, the y-intercept is at point (0, 48).
c) Determine the minimum or maximum value of the graph of the function.
Since the function is a polynomial, there is no restriction on the values it can take. Therefore, there is no minimum or maximum value for this function.
d) Describe the end behavior of the graph of the function.
As described in part (a), the end behavior of the graph is as follows:
- As x approaches positive infinity, the function decreases without bound, approaching negative infinity.
- As x approaches negative infinity, the function increases without bound, approaching positive infinity.
e) Sketch the graph of the function using the information obtained in parts (a) through (d).
The graph of the function will have a similar shape to a cubic function with a negative leading coefficient. It will pass through the x-intercepts of -2 and 2 and the y-intercept of (0, 48). As x approaches positive infinity, the graph will decrease without bound, and as x approaches negative infinity, the graph will increase without bound.