Final answer:
To sketch the region between the curves y = |6x| and y = x² - 7, plot the individual shapes of each curve, their points of intersection, and shade the area between them where they overlap.
Step-by-step explanation:
To sketch the region enclosed by the curves y = |6x| and y = x² - 7, we start by understanding the shape of each equation on a graph. The absolute value function y = |6x| splits into two lines: y = 6x for x ≥ 0 and y = -6x for x < 0, both passing through the origin and forming a V-shape. The quadratic equation y = x² - 7 is a parabola that opens upwards with the vertex at the point (0, -7).
First, we find the intersection points of the two curves. Setting these two equations equal to each other, and solving for x, gives the intersections points. There will be two points: one in the first quadrant and one in the second quadrant where the V-shaped graph intersects the parabola.
Once the points of intersection are determined, we can sketch the curves and shade the enclosed region. It's important to carefully consider each branch of the V-shaped graph, as the absolute value function affects the positive and negative x-values differently.