SOLUTION: Find the quadratic equation that fits the following set of data points. (−2​,−8​),(−4​,−12​),(1​,13​) ​f(x)=____

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Question 1114193: Find the quadratic equation that fits the following set of data points.
(−2​,−8​),(−4​,−12​),(1​,13​)
​f(x)=______.
​(Use integers or fractions for any numbers in the expression. Do not​ factor.)
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
the standard form of your quadratic equation is:

y = ax^2 + bx + c

when x = -2 and y = -8, this equation becomes:

-8 = 4a -2b + c

when x = -4, this equation becomes:

-12 = 16a - 4b + c

when x = 1, this equation becomes:

13 = a + b + c

you have 3 equations thst need to be solved simultaneously.

they are:

-8 = 4a - 2b + c
-12 = 16a - 4b + c
13 = a + b + c

you can reorder the terms to get:

4a - 2b + c = -8
16a - 4b + c = -12
a + b + c = 13

they are now in standard linear equation form.

when you solve these equations simultaneously, you will find that:

a = 1
b = 8
c = 4

since the standard form of your equation is y = ax^2 + bx + c, you get:

y = x^2 + 8x + 4

that's your quadratic equation.

when x = -2, you get y = (-2)^2 - 8*2 + 4 = -8

when x = -4, you get y = (-4)^2 - 8*4 + 4 = -12

when x = 1, you get y = 1^2 - 8*1 + 4 = 13

this confirms your equation is correct.

i'm assuming you know how to solve 3 equations in 3 unknowns simultaneously.

i solved as follows.
reference the worksheet shown below.

$$$

i started with equations 1, 2, and 3.
those are the circled numbers on the worksheet.

i then took equation 2 as is and multiplied equation 1 by 4 to get equation 4.

i then subtracted equation 4 from equation 2 to get equation 5.

i then took equation 2 as is and multiplied equation 3 by 16 to get equation 6.

i then subtracted equation 6 from equation 2 to get equation 7.

i then copied down equation 5 and 7 and proceeded to reduced these further so that i could solve for a single variable.

i multiplied equation 5 by 5 to get equation 8 and kept equation 7 as is.

i then subtracted equation 7 from equation 8 to get equation 9.

i then solved equation 9 for c to get c = 4

i then took equation 5 and replaced c with 4 and solved for b to get b = 8.

i then took equation 1 and replaced c with 4 and b with 8 and solved for a to get a = 1.

the result was a = 1, b = 8, c = 4.

that led to y = x^2 + 8x + 4, which is the solution shown above.

















Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Find the quadratic equation that fits the following set of data points.
(−2​,−8​),(−4​,−12​),(1​,13​)
​f(x)=______.
​(Use integers or fractions for any numbers in the expression. Do not​ factor.)
Standard equation of a quadratic function: 

(− 2​,− 8​)

 ------ Substituting (- 2, - 8) for (x, y)
4a - 2b + c = - 8 ------- eq (i)


(− 4​, − 12​)

 ------ Substituting (- 4, - 12) for (x, y)
16a - 4b + c = - 12 ------- eq (ii)


(1​, 13​)

 ------ Substituting (1, 13) for (x, y)
a + b + c = 13 ------- eq (iii)

 4a - 2b + c = -  8 ------- eq (i)
16a - 4b + c = - 12 ------- eq (ii)
  a +  b + c =   13 ------- eq (iii)

12a - 2b = - 4 ------ Subtracting eq (i) from eq (ii) 
2(6a - b) = 2(- 2) 
6a - b = - 2 ------ eq (iv)

3a - 3b = - 21 ------ Subtracting eq (iii) from eq (i) 
3(a - b) = 3(- 7) 
a - b = - 7 ------ eq (v)

5a = 5 ------- Subtracting eq (v) from eq (iv)


1 - b = - 7 ------- Substituting 1 for a in eq (v)
- b = - 7 - 1
- b = - 8



1 + 8 + c = 13 ------- Substituting 1 for a, and 8 for b in eq (iii)
9 + c = 13




 ------- Substituting 1 for a, 8 for b, and 4 for c
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