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.
(a) Find the x-intercepts of the function.
To find the x-intercepts, set f(x) = 0 and solve for x:
0 = x^2(x-6)
0 = x^2 or 0 = x-6
This gives x = 0 or x = 6.
Therefore, the x-intercepts are at x = 0 and x = 6.
(b) Find the y-intercept of the function.
To find the y-intercept, set x = 0:
f(0) = 0^2(0-6) = 0
Therefore, the y-intercept is at (0,0).
(c) Find the end behavior of the function.
As x approaches positive or negative infinity, the leading term of the function will dominate. Since the leading term is x^3, the end behavior of the function is that it increases without bound as x approaches positive infinity and decreases without bound as x approaches negative infinity.
(d) Find the axis of symmetry of the function.
The axis of symmetry for a quadratic function in the form f(x) = ax^2 + bx + c is given by x = -b/(2a). In this case, a = 1 and b = 0.
Therefore, the axis of symmetry is x = 0.
(e) Find the vertex of the function.
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by the point (h, k), where h = -b/(2a) and k = f(h).
In this case, h = 0 and k = f(0) = 0.
Therefore, the vertex of the function is at (0,0).
By analyzing the x-intercepts, y-intercept, end behavior, axis of symmetry, and vertex, we can sketch a complete graph of the function f(x) = x^2(x-6). We know that the function passes through the points (0,0) and (6,0), is symmetric about the y-axis, and has an end behavior that increases without bound as x approaches positive infinity and decreases without bound as x approaches negative infinity.