Question

Sketch a graph of the polynomial function f(x) = –x³ + 5x² – 2x – 8. Use the graph to complete the following:

f is __________ on the interval (1/3,3)

Fill in the above blank with one of the following options:
A. positive
B. negative
C. decreasing
D. increasing

Answer 1

To sketch the graph of the polynomial function f(x) = –x³ + 5x² – 2x – 8, find the x-intercepts, y-intercept, test values to determine behavior, and sketch the curve. The function f(x) is negative on the interval (1/3,3).

Step-by-step explanation:

To sketch the graph of the polynomial function f(x) = –x³ + 5x² – 2x – 8, we can follow these steps:

1. Find the x-intercepts by setting f(x) equal to zero and solving for x.
2. Find the y-intercept by evaluating f(0).
3. Determine the behavior of the function at the x-intercepts by testing a value to the left and right of each x-intercept.
4. Plot the x-intercepts, y-intercept, and points that represent the behavior at the x-intercepts.
5. Sketch the curve so that it passes through the plotted points and follows the general shape of a third-degree polynomial function.

The graph of f(x) = –x³ + 5x² – 2x – 8 will be a curve that passes through the x-intercepts, has a y-intercept, and follows the general shape of a cubic function. To determine whether f is positive or negative on the interval (1/3,3), we can test any value between 1/3 and 3 in the function. For example, we can choose x = 1. Plug it into the function: f(1) = –1³ + 5(1)² – 2(1) – 8 = -1 + 5 – 2 – 8 = -6. Since f(1) is negative, the function f(x) is negative on the interval (1/3,3).