Sketching the Region Enclosed by the Curves
To sketch the region enclosed by the curves defined by the equations y = 9x and y = x² + 10, we will follow these steps:
1. Identify the Curves
The first curve is a linear function, y = 9x, which is a straight line with a slope of 9 passing through the origin. The second curve, y = x² + 10, is a parabolic function that opens upwards, with its vertex at (0, 10).
2. Find the Points of Intersection
To find the bounded region, we need to determine where these two curves intersect. To do this, we set them equal to each other:
9x = x² + 10Rearranging this equation gives:
x² - 9x + 10 = 0This can be factored into:
(x - 5)(x - 2) = 0Thus, the points of intersection are:
x = 5 and x = 23. Sketching the Curves
Now that we know the points of intersection, we can sketch the graphs of both functions.
- For y = 9x: Start from the origin (0,0), and as you move to the right, the line steeply rises.
- For y = x² + 10: This curve starts at the vertex (0, 10) and opens upward. The points of intersection (2, 28) and (5, 55) should be marked on this parabola.
4. Determine the Bounded Region
The enclosed region is where y = 9x lies below y = x² + 10. For the interval between the points of intersection (from x = 2 to x = 5), the curve y = x² + 10 is always above y = 9x.
5. Final Sketch
The final sketch should display the straight line y = 9x and the parabola y = x² + 10, clearly marking the intersection points at (2, 18) and (5, 55). The area between these two curves represents the region that is enclosed.
When creating your sketch, consider adding shading between the curves to highlight the enclosed area.