Final answer:
The graph of the cubic polynomial function f(x) = x^3 - 15x^2 + 68x - 96 demonstrates an S-shaped curve with the end behavior showing as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity.
Step-by-step explanation:
The graph of the function f(x) = x3 − 15x2 + 68x − 96 is a cubic polynomial, which typically has the shape of an 'S' or an inverted 'S' depending on the sign of the leading coefficient. As this is a function with the highest power of x being 3 and a positive leading coefficient, the end behavior is such that as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity. At x = 0, the function has a y-intercept at f(0) = -96. The shape of the graph changes as the constants are adjusted, and each term contributes to the overall shape of the polynomial curve. Labeling the graph for this function should include an appropriate scale for the x-axis and y-axis, with x ranging from negative to positive values that reveal the behavior of the curve, and f(x) scaled to show the critical points on the graph.
End Behavior
The following describes the end behavior of the function:
- As x → −∞, f(x) → −∞
- As x → ∞, f(x) → ∞