find the quadratic equation that fits the following set of data points (5,-238),(-3,-110)(4,152)

Question

I have tried doing this problem over 10 times. I can not figure it out :( please help

Answer 1

Here's how to set up the problem. Start by writing a quadratic equation of the form y=ax²+bx+c for each of the x,y points you are given:

 

For the point (5,-238) the equation is: -238 = a(5)² + b(5) + c

For the point (-3,-110) the equation is: -110 = a(-3)² + b(-3) + c

For the point (4,152) the equation is: 152 = a(4)² + b(2) + c

 

Expanding the exponent you get a system of three equations, which I will number 1 through 3: 

equation 1: -238 = 25a + 5b + c

equation 2: -110 =   9a -  3b + c

equation 3:  152 = 16a + 2b + c

 

You now have a system of three equations with three unknowns (a, b, and c). We can subtract equation 2 from equation 1 to get a new equation with a and b:

equation 1 - equation 2 = -238 - (-110) = (25a - 9a) + (5b - (-3b)) + (c - c)

Let's call this new equation equation 4: 

 

equation 4: -128 = 16a + 8b

 

Do the same thing with equations 2 and 3 - subtract equation 3 from equation 2 to get a new equation (call it equation 5) that only has a and b in it. 

 

Once you have the two equations with only a and b, you can combine those to solve for a and b. Once you've solved for a and b, plug a and b into any of the original equations (1 through 3) and solve for c. 

Answer 2

The target equation: y = ax² + bx + c

 

Use the points to get a system of equations:

 

25a + 5b + c = -238

  9a -  3b + c = -110

16a + 4b + c = 152

 

Subtract the second equation from the first and divide by 8. Also, subtract the third equation from the first. This gives a new system.

 

2a + b = -16

9a + b = -390

 

In this system, subtract the first equation from the second.

 

7a = -374

 

a = -374/7

 

b = -16 - 2a = -112/7 + 748/7 = 636/7

 

c = -110 - 9a + 3b = -770/7 + 3366/7 + 1908/7 = 4504/7

 

Thus:

 

y = (-374x² + 636x + 4504)/7