The vertex of the parabola below is at the point 6 1.5 and the point 7 3.5 is on the parabola What is the equation of the parabola?

Answer 1

Since the vertex of the parabola is at the point (6, 1.5) and the point (7, 3.5) is on the parabola, the point (5, 3.5) also is on the parabola (since the axis of symmetry is x = 6). So we have:

y = ax2 + bx + c (the general equation of a parabola)

(5, 3.5); 3.5 = a(5)2 + b(5) + c

3.5 = 25a + 5b + c (1)

(7, 3.5); 3.5 = a(7)2 + b(7) + c

3.5 = 49a + 7b + c (2)

(6, 1.5); 1.5 = a(6)2 + b(6) + c

1.5 = 36a + 6b + c (3)

Subtract the equation (1) and (3) from the equation (2).

0 = 24a + 2b

2 = 13a + b which yields b = 2 - 13a

0 = 24a + 2b (substitute 2 - 13a for b)

0 = 24a + 2(2 - 13a)

0 = 24a + 4 - 26a

0 = -2a + 4

-4 = -2a

2 = a

b = 2 - 13a (substitute 2 for a)

b = 2 - 13(2) = 2 - 26 = -24

3.5 = 25a + 5b + c (substitute 2 and -24 for a and b)

3.5 = 25(2) + 5(-24) + c

3.5 = 50 - 120 + c

73.5 = c

y = ax2 + bx + c

y = (2)x2 + (-24)x + 73.5

y = 2x2 - 24x + 73.5 (the equation of the parabola)

Or,

vertex = (6, 1.5) = (-b/2a, c - b2/4a);

6 = -b/2a

b = -12a

1.5 = c - b2/4a

c = 1.5 + b2/4a = 1.5 + (-12a)2/4a = 1.5 + 144a2/4a = 1.5 + 36a

y = ax2 + bx + c

3.5 = a(7)2 + b(7) + c

3.5 = 49a + 7b + c

3.5 = 49a + 7(-12a) + 1.5 - 36a

3.5 = 49a - 84a + 1.5 + 36a

3.5 = a + 1.5

2 = a

b = -12a = -12(2) = -24

c = 1.5 + 36a = 1.5 + 36(2) = 1.5 + 72 = 73.5