Graph the polynomial function f(x)=-2(x-1)^2(x^2-9)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|
(a) As x approaches positive infinity, the leading term of the polynomial function is -2(x-1)^2(x^2), which simplifies to -2x^5. Therefore, the end behavior of the graph is y = -2x^5 as x approaches positive infinity.
(b) Find the x-intercepts of the graph of the function.
To find the x-intercepts, we set f(x) = 0:
-2(x-1)^2(x^2-9) = 0
This equation is true when x = 1 or x = -3 or x = 3. Therefore, the x-intercepts are at x = 1, x = -3, and x = 3.
(c) Find the y-intercept of the graph of the function.
To find the y-intercept, we set x = 0:
f(0) = -2(0-1)^2(0^2-9) = -18
Therefore, the y-intercept is at y = -18.
(d) Determine the multiplicity of each x-intercept.
The x-intercepts are at x = 1, x = -3, and x = 3. The multiplicity of each x-intercept can be determined by looking at the corresponding factor in the function.
At x = 1, the factor is (x-1)^2, so the multiplicity is 2.
At x = -3 and x = 3, the factors are (x+3) and (x-3), respectively, so the multiplicity for each is 1.
(e) Sketch the graph of the function.
The graph will have the x-intercepts at x = 1, x = -3, and x = 3 with multiplicities 2, 1, and 1, respectively. The y-intercept is at y = -18. The end behavior of the graph is y = -2x^5. The graph will also open downwards since the leading coefficient is negative.