how do you find the equation of a parabola given three points it passes through?

The coordinates of all three points must satisfy the equation y = ax² + bx = c.

For (- 2, 18):

x = - 2

y = 18

18 = a(- 2)² + b(- 2) + c

18 = a(4) - 2b + c

18 = 4a - 2b + c

4a - 2b + c = 18 .......................... Eq. 1

For (1, 3):

x = 1

y = 3

3 = a(1)² + b(1) + c

3 = a + b + c

a + b + c = 3 .............................. Eq. 2

For (5, 39):

x = 5

y = 39

39 = a(5)² + b(5) + c

39 = a(25) + 5b + c

39 = 25a + 5b + c

25a + 5b + c = 39 ....................... Eq. 3

Subtract Eq. 2 from Eq. 1:

4a - a - 2b - b + c - c = 18 - 3

3a - 3b = 15

a - b = 5

b = a - 5 ..................................... Eq. 4

Subtract Eq. 2 from Eq. 3:

25a - a + 5b - b + c - c = 39 - 3

24a + 4b = 36

6a + b = 9 .................................. Eq. 5

Sub a - 5 from Eq. 4 for b in Eq. 5:

6a + (a - 5) = 9

6a + a = 9 + 5

7a = 14

a = 14 / 7

a = 2

Sub 2 for a in Eq. 4:

b = 2 - 5

b = - 3

Sub 2 for a and - 3 for b in Eq. 2:

2 - 3 + c = 3

- 1 + c = 3

c = 3 + 1

c = 4

Equation of Parabola:

y = 2x² - 3x + 4

¯¯¯¯¯¯¯¯¯¯¯¯¯

Evey parabola's equation is like y=ax^2 +bx+c so if it passes from these points they should verify the parabola if ypu put them in x and y so: 18=4a-2b+c

3=a+b+c

39=25a+5b+c and if you solve these three equations as a system you'll find a,b,c and you can replace them in an equation

A unique solution is not possible with only three points. Do you have any other information? The orientation of the axis would do it. In standard form the axis is usually horizontal. Quite often y is expressed as a quadratic function of x, in which case the axis would be vertical. If it is in general form, then the axis could be oriented in any direction.