Final answer:
The function F(x) has cubic end behaviors; as x approaches infinity or negative infinity, the function's value does the same. The y-intercept is at the origin (0,0), and the x-intercepts are located at -6, -2, and 0, where the function crosses the x-axis.
Step-by-step explanation:
The cubic function F(x)=x³+8x²+12x can be graphed to analyze its end behaviors, y-intercept, and x-intercepts. To determine these characteristics, first, let's consider the end behaviors. For large positive or large negative values of x, the term x³ dominates the behavior of the function, so as x → ∞, F(x) → ∞ and as x → -∞, F(x) → -∞. This implies that the graph goes up on both ends.
Next, we can find the y-intercept by setting x = 0, which gives us F(0) = 0³ + 8*0² + 12*0 = 0. So, the y-intercept is at (0, 0).
For the x-intercepts, we set F(x) to zero and solve for x. The equation simplifies to x(x²+8x+12) = 0. Factoring the quadratic part, we get x(x+2)(x+6) = 0, which gives us three x-intercepts at -6, -2, and 0. These are the points where the graph crosses the x-axis.