Sketch the graph and find the area of the region bounded below by the graph of the given function and above by the x-axis from x = a to x = b.

Question

Quizplus · Accepted Answer

The graph of the function y = x^2 - 2x - 8 is a parabola that opens upwards. To find the x-intercepts, we set y = 0 and solve for x:
0 = x^2 - 2x - 8
0 = (x - 4)(x + 2)
x = 4 or x = -2
Therefore, the graph intersects the x-axis at these points.

To find the area of the region bounded by the graph and the x-axis from x = a to x = b, we need to integrate the function from x = a to x = b:
A = ∫(b to a) (x^2 - 2x - 8) dx
A = [x^3/3 - x^2 - 8x]b^a
A = [(b^3 - a^3)/3 - (b^2 - a^2) - 8(b - a)]

Using these formulas and plugging in a = -2 and b = 4, we get:
A = [(4^3 - (-2)^3)/3 - (4^2 - (-2)^2) - 8(4 - (-2))]
A = 14

Therefore, the area of the region is 14 square units. The graph and the shaded region are shown in choice C.

Sketch the Graph and Find the Area of the Region