Parabola Question

Question

How do I find the equation of this parabola y = ax² + bx + c

that passes through the following points: (0,3), (1,4), (2,3) ?

Answer 1

Coordinate (0,3) --> when x = 0, y = 3

Substitute 0 for x and 3 for y:

3 = a(0)^2 + b(0) + c

c = 3

When x = 1, y = 4

Substitute 1 for x and 4 for y and 3 for c:

4 = a(1)^2 + (1)b + 3

4 = a + b + 3

a + b = 1

a = 1 - b

When x =2, y = 3

Substitute 2 for x and 3 for y and 3 for c:

3 = a(2)^2 + (2)b + 3

3 = 4a + 2b + 3

0 = 4a + 2b

substitute 1-b for a in above equation and solve for b:

0 = 4(1-b) + 2b

0 = 4 - 4b + 2b

-4 = -2b

b = 2

a = 1 - b = 1 - 2 = -1

so a = -1, b = 2, c = 3

equation of parabola is: y = -x^2 + 2x + 3

Answer 2

Hi Lily! Thanks for asking a good question!

Notice that the three coordinates are all x and y values.

Substitute each x and y value into the equation three separate times.

With (0, 3), 3 = a * (0)^2 + b * (0) + c = c = 3

With (1, 4), 4 = a * (1)^2 + b * (1) + (3) = a + b + 3 = 4

With (2, 3), 3 = a * (2)^2 + b * (2) + (3) = 4a + 2b + 3 = 3

Therefore, with the last two equations,

a + b = 1

4a + 2b = 0.

Multiply by -2 on the top equation on both sides (there's more than one way to solve this) to get

-2a + -2b = -2

4a + 2b = 0

Notice the 2b's cancel each other. Therefore, add -2a and 4a on one side, and add -2 and 0 on the other side to get:

2a = -2 or a = -1

Using the equation (of course, you could also use the other one) a + b = 1, b = 2.

Therefore, y = -1 * x^2 + 2x + 3 ---------- (ANS)

Hope this helps :)