Use a system of equations to find the parabola of the form y=ax2+bx+c that goes through the three points given. (2,2), (-4,50), (3,8) y=?

 We must form the system of equations. We will do that by replacing variables `x` and `y` in equation of parabola `y=ax^2+bx+c` with coordinates of the given points. Thus, we get system of 3 equations with 3 unknowns `a,b` and `c.`

`2=2^2a+2b+c`

` ` `50=(-4)^2a-4b+c`

`8=3^2a+3b+c`

`4a+2b+c=2`

`16a-4b+c=50`

`9a+3b+c=8`

There are several ways to solve this system of equations. We will use Gaussian elimination.

Subtract first equations from the second and then from the third.

`4a+2b+c=2`

`12a-6b=48`

`5a+b=6`

Add third equation multiplied...

 We must form the system of equations. We will do that by replacing variables `x` and `y` in equation of parabola `y=ax^2+bx+c` with coordinates of the given points. Thus, we get system of 3 equations with 3 unknowns `a,b` and `c.`

`2=2^2a+2b+c`

` ` `50=(-4)^2a-4b+c`

`8=3^2a+3b+c`

`4a+2b+c=2`

`16a-4b+c=50`

`9a+3b+c=8`

There are several ways to solve this system of equations. We will use Gaussian elimination.

Subtract first equations from the second and then from the third.

`4a+2b+c=2`

`12a-6b=48`

`5a+b=6`

Add third equation multiplied by 6 to the second equation.

`4a+2b+c=2`

`42a=84`

`5a+b=6`

Now we solve second equation, then third and then first.

`42a=84`

`a=2`

Now we use the value of `a` to solve the third equation.

`5cdot2+b=6`

`10+b=6`

`b=-4`

Now we use values of `a` and `b` to solve the first equation.

`4cdot2+2cdot(-4)+c=2`

`8-8+c=2`

`c=2`

Now that we know values of all three parameters `a,b` and `c,` we can write the equation of parabola.

`y=2x^2-4x+2`         

Graph of the parabola can be seen in the image below.