Answer:
Explanation:
To sketch the graph of the polynomial function P(x) = x^4 - x^3 - 11x^2 + 9x + 18 without using technology, we can follow the following steps:
1. Determine the end behaviors of the graph: As x approaches negative infinity, the dominant term of the polynomial is x^4, which is positive. As x approaches positive infinity, the dominant term of the polynomial is also x^4, which is positive. Therefore, the end behaviors of the graph are both upward, which means that the graph rises to positive infinity on both ends.
2. Find the x-intercepts: To find the x-intercepts, we need to solve the equation P(x) = 0. One way to do this is to use synthetic division or long division to factor the polynomial. However, since the degree of the polynomial is 4, it may be difficult to find the roots algebraically. Alternatively, we can use technology or a graphing calculator to find the approximate values of the x-intercepts. Using a graphing calculator, we find that the x-intercepts are approximately -2.23, -0.43, 1.19, and 3.47.
3. Find the y-intercept: To find the y-intercept, we can set x = 0 in the equation P(x) = x^4 - x^3 - 11x^2 + 9x + 18, which gives P(0) = 18. Therefore, the y-intercept is (0, 18).
4. Determine the location of the minimum point: To find the location of the minimum point, we can use calculus and take the derivative of P(x) with respect to x. Setting the derivative equal to zero and solving for x, we find that x = -0.5 or x = 1. Therefore, the minimum point occurs at either (-0.5, P(-0.5)) or (1, P(1)). Using a graphing calculator, we find that the minimum value of P(x) is approximately -8.09, which occurs at x = -0.5.
Based on these steps, we can sketch the graph of P(x) as follows:
* The end behaviors of the graph are both upward.
* The x-intercepts are approximately -2.23, -0.43, 1.19, and 3.47.
* The y-intercept is (0, 18).
* The minimum point occurs at approximately (-0.5, -8.09).
* The graph is symmetric about the vertical line x = 0, since P(-x) = P(x) for all x.
5. To find where the company will break even, we need to find the values of x such that P(x) = 0. Using a graphing calculator or technology, we find that the company will break even at approximately x = -2.23, -0.43, 1.19, and 3.47. These correspond to the x-intercepts of the graph. Therefore, the company will break even at these values of x.