The solution set of the inequality y(>/) 1/3(x-4)^2+1 includes the points
A. (–1, 1), (4, 0), and (7, –1)
B. (1, 3), (4, 1), and (8, 2)
C. (4, 2), (7, 4), and (8, 8)
D. (–4, 2), (0, 9), and (9, 1)
To determine the solution set of the inequality y ≥ 1/3(x-4)^2+1, we need to substitute the given points into the inequality and check if they satisfy the inequality.
Let's evaluate each option:
Option A:
- Point (-1, 1):
y ≥ 1/3((-1)-4)^2+1
1 ≥ 1/3(5)^2+1
1 ≥ 1/3(25)+1
1 ≥ 25/3+1
1 ≥ 8.33
The point (-1, 1) does not satisfy the inequality.
- Point (4, 0):
y ≥ 1/3((4)-4)^2+1
0 ≥ 1/3(0)^2+1
0 ≥ 0+1
0 ≥ 1
The point (4, 0) does not satisfy the inequality.
- Point (7, -1):
y ≥ 1/3((7)-4)^2+1
-1 ≥ 1/3(3)^2+1
-1 ≥ 1/3(9)+1
-1 ≥ 3+1
-1 ≥ 4
The point (7, -1) does not satisfy the inequality.
Option A does not include any points that satisfy the inequality.
Option B:
- Point (1, 3):
y ≥ 1/3((1)-4)^2+1
3 ≥ 1/3(-3)^2+1
3 ≥ 1/3(9)+1
3 ≥ 3+1
3 ≥ 4
The point (1, 3) does not satisfy the inequality.
- Point (4, 1):
y ≥ 1/3((4)-4)^2+1
1 ≥ 1/3(0)^2+1
1 ≥ 1/3(0)+1
1 ≥ 0+1
1 ≥ 1
The point (4, 1) satisfies the inequality.
- Point (8, 2):
y ≥ 1/3((8)-4)^2+1
2 ≥ 1/3(4)^2+1
2 ≥ 1/3(16)+1
2 ≥ 16/3+1
2 ≥ 5.33
The point (8, 2) satisfies the inequality.
Option B includes the point (4, 1) and (8, 2) that satisfy the inequality.
Option C:
- Point (4, 2):
y ≥ 1/3((4)-4)^2+1
2 ≥ 1/3(0)^2+1
2 ≥ 1/3(0)+1
2 ≥ 0+1
2 ≥ 1
The point (4, 2) satisfies the inequality.
- Point (7, 4):
y ≥ 1/3((7)-4)^2+1
4 ≥ 1/3(3)^2+1
4 ≥ 1/3(9)+1
4 ≥ 3+1
4 ≥ 4
The point (7, 4) satisfies the inequality.
- Point (8, 8):
y ≥ 1/3((8)-4)^2+1
8 ≥ 1/3(4)^2+1
8 ≥ 1/3(16)+1
8 ≥ 16/3+1
8 ≥ 5.33
The point (8, 8) satisfies the inequality.
Option C includes the points (4, 2), (7, 4), and (8, 8) that satisfy the inequality.
Option D:
- Point (-4, 2):
y ≥ 1/3((-4)-4)^2+1
2 ≥ 1/3(-8)^2+1
2 ≥ 1/3(64)+1
2 ≥ 64/3+1
2 ≥ 22.33
The point (-4, 2) satisfies the inequality.
- Point (0, 9):
y ≥ 1/3((0)-4)^2+1
9 ≥ 1/3(-4)^2+1
9 ≥ 1/3(16)+1
9 ≥ 16/3+1
9 ≥ 5.33+1
9 ≥ 6.33
The point (0, 9) satisfies the inequality.
- Point (9, 1):
y ≥ 1/3((9)-4)^2+1
1 ≥ 1/3(5)^2+1
1 ≥ 1/3(25)+1
1 ≥ 25/3+1
1 ≥ 8.33+1
1 ≥ 9.33
The point (9, 1) does not satisfy the inequality.
Option D includes the points (-4, 2) and (0, 9) that satisfy the inequality.
Thus, the correct answer is Option D. The solution set of the inequality y ≥ 1/3(x-4)^2+1 includes the points (-4, 2) and (0, 9).
it's easy to check x=0. That makes y>= 19/3
check the other points in like manner
the vertex is at (4,1) so y will be greater than 1 in all cases