Question

What is the maximum vertical distance between the line y = x + 42 and the parabola y = x² for −6 ≤ x ≤ 7?

Answer 1

The maximum vertical distance between the line y = x + 42 and the parabola y = x² for -6 ≤ x ≤ 7 is 42.

Step-by-step explanation:

To find the maximum vertical distance between the line y = x + 42 and the parabola y = x² for -6 ≤ x ≤ 7, we need to find the highest point on the parabola that is above or below the line. We can do this by finding the vertex of the parabola and comparing its y-coordinate with the y-coordinate of the line at that x-value.

The vertex of the parabola y = x² is the point (h, k), where h is given by the formula h = -b/(2a) and k is the y-coordinate of the vertex. In this case, a = 1 and b = 0, so h = 0 and k = 0. This means that the vertex of the parabola is at the point (0, 0).

Substituting x = 0 into the equation of the line y = x + 42, we get y = 0 + 42 = 42. Therefore, the maximum vertical distance between the line y = x + 42 and the parabola y = x² is 42.

Answer 2

To calculate the maximum vertical distance between the line y = x + 42 and the parabola y = x² over the interval −6 ≤ x ≤ 7, we find the maximum value of |x + 42 - x²| either by evaluating the expression at key points or using calculus to find critical points.

Step-by-step explanation:

To find the maximum vertical distance between the line y = x + 42 and the parabola y = x² over the interval from −6 ≤ x ≤ 7, we need to calculate the difference in y-values that the line and the parabola have for any given x within this range. The difference at any point x is given by the formula vertical distance = |y(line) - y(parabola)| or |x + 42 - x²|.

To find the maximum value of this expression over the interval, we could take derivatives and find the critical points or directly evaluate the expression at the endpoints and any local extremes within the interval. Since this is a high school level question, using derivatives might not be applicable unless the student is familiar with calculus.

After evaluating these values over the specified interval, the maximum vertical distance is the largest difference calculated. This would provide you with the maximum vertical distance between the line and the parabola.