Analyze the polynomial function f(x)=x^2(x-6) using parts (a) through (e). Question content area bottom Use the above information to draw a complete graph of the function. Choose the correct graph below.
(a) Determine the degree of the polynomial: The degree of the polynomial function f(x) is the highest power of x in the function. In this case, the highest power of x is 3 (x^2 * x = x^3), so the degree of the polynomial is 3.
(b) Determine the leading coefficient: The leading coefficient is the coefficient of the term with the highest power of x. In this case, the leading coefficient is 1 (from x^2), so the leading coefficient is 1.
(c) Find the x-intercepts: To find the x-intercepts, set f(x) equal to 0 and solve for x:
x^2(x-6) = 0
x^2 = 0 or x-6 = 0
x = 0 or x = 6
So the x-intercepts are at x = 0 and x = 6.
(d) Find the y-intercept: To find the y-intercept, set x = 0 in the function f(x):
f(0) = 0^2(0-6) = 0
So the y-intercept is at y = 0.
(e) Determine the end behavior: The end behavior of the function is determined by the leading term, which in this case is x^3. Since the degree is odd and the leading coefficient is positive, the end behavior will be as x approaches positive or negative infinity, f(x) will also approach positive or negative infinity.
Based on the information above, the graph of the function f(x) = x^2(x-6) will have x-intercepts at x = 0 and x = 6, a y-intercept at y = 0, a degree of 3, a leading coefficient of 1, and end behavior indicated as f(x) approaches positive or negative infinity. The correct graph will show a polynomial function with these characteristics.