Final answer:
The maximum vertical distance between the line y = x + 6 and the parabola y = x^2 is 5 units.
Step-by-step explanation:
To find the maximum vertical distance between the line y = x + 6 and the parabola y = x2 for -2 ≤ x ≤ 3, we need to determine the points of intersection between the line and the parabola. To do this, we set the equations equal to each other:
x + 6 = x2
Now we have a quadratic equation. Rearranging and setting it equal to zero:
x2 - x - 6 = 0
Factoring the quadratic equation:
(x - 3)(x + 2) = 0
Setting each factor equal to zero and solving for x:
x - 3 = 0 or x + 2 = 0
x = 3 or x = -2
So the points of intersection are (-2, 4) and (3, 9).
The maximum vertical distance will occur at one of these points. To find the vertical distances, we substitute the x-values into each equation:
For x = -2: y = (-2) + 6 = 4
For x = 3: y = (3) + 6 = 9
Therefore, the maximum vertical distance is 9 - 4 = 5 units.