Final answer:
To see all the intercepts and the maximum of the function y = -x^2 + 3x + 207, the minimum window boundary for the x-axis should be around x = -17.53 and the maximum window boundary for the x-axis should be around x = 11.53. The minimum window boundary for the y-axis should be around y = 441/4, and the maximum window boundary for the y-axis should also be around y = 441/4.
Step-by-step explanation:
In order to see all the intercepts and the maximum for the function y = -x^2 + 3x + 207, we need to determine the minimum and maximum window boundaries for the x-axis and y-axis. The coefficient of the x^2 term is negative, which means the parabola opens downward. To find the x-coordinate of the maximum point, we use the formula x = -b/(2a), where a is the coefficient of the x^2 term and b is the coefficient of the x term. Plugging in the values from the function, we get x = -3/(2*(-1)) = -3/(-2) = 3/2.
To determine the minimum and maximum boundaries for the x-axis, we need to consider the x-intercepts. We can find the x-intercepts by setting y = 0 and solving for x. Solving -x^2 + 3x + 207 = 0, we can use factoring or the quadratic formula to find the x-intercepts. The x-intercepts are approximately x = 11.53 and x = -17.53. Therefore, the minimum window boundary for the x-axis should be around x = -17.53 and the maximum window boundary should be around x = 11.53.
The minimum window boundary for the y-axis should be the minimum value of y, which occurs at the maximum point. Plugging x = 3/2 into the function, we get y = -(3/2)^2 + 3(3/2) + 207 = -9/4 + 9/2 + 207 = 441/4. Therefore, the minimum window boundary for the y-axis should be around y = 441/4.
The maximum window boundary for the y-axis should be the maximum value of y, which occurs at the intercept with the x-axis closest to the maximum point. Since the x-coordinate of the maximum point is 3/2, we can use this value to find the maximum value of y. Plugging x = 3/2 into the function, we get y = -(3/2)^2 + 3(3/2) + 207 = -9/4 + 9/2 + 207 = 441/4. Therefore, the maximum window boundary for the y-axis should also be around y = 441/4.
Learn more about finding window boundaries for a function