Find the equation of the parabola that passes through the points (-2,24), (3,-1), and (-1,15)

Question

i have been stuck on this question for a very long time and i need advice on how to solve it

Answer 1

Basic parabola:  y = ax² + bx + c

We have 3 points we can plug in for (x, y) to create 3 simultaneous equations

 

(-2, 24):  24 = 4a - 2b + c   {equation 1}

(3, -1):   -1 = 9a + 3b + c    {equation 2}

(-1, 15): 15 = a - b + c        {equation 3}

 

Solve this system to find the values of a, b, c

 

Let's first eliminate variable c:

 

4a - 2b + c = 24   {equation 1}

  a -  b + c = 15    {equation 3}

---------------------  subtract

3a - b = 9

 

9a + 3b + c = -1    {equation 2}

  a -  b + c = 15    {equation 3}

--------------------  subtract

8a + 4b = -16

 

We now have two equations with 2 unknowns we can use to find a, b

 

3a - b = 9         {equation 4}

8a + 4b = -16   {equation 5}

 

Multiply equation 4 through by 4 and add equations

 

12a - 4b = 36

8a + 4b = -6

-----------------   add

20a = 30

a = 30/20

a = 3/2

 

8a + 4b = -6

8(3/2) + 4b = -6

12 + 4b = -6

4b = -6- 12

4b = -18

b = -18/4

b = -9/2

 

Plug these 2 values into one of the original equations and solve for c

 

15 = a - b + c        {equation 3}

15 = 3/2 + 9/2 + c

15 = 12/2 + c

15 = 6 + c

c = 15-6

c = 9

 

y = (3/2)x² - (9/2)x+ 9