A quadratic function is a function of the form y=ax^2+bx+c where a, b, and c are constants. Given any 3 points in the plane, there is exactly one quadratic function whose graph contains these points. Find the quadratic function whose graph contains the points (0, -2), (-5, -17), and (3, -17). Enter the equation below. Function: y = 0

We know that a quadratic function is a function of the form [tex]y=ax^{2}+bx+c[/tex]. The first step is use the 3 points given to write 3 equations to find the values of the constants a,b, and c.

Substitute the points (0,-2), (-5,-17), and (3,-17) into the general form of a quadratic function.

[tex]-2=a*0^{2}+b*0+c\\c=-2[/tex]

[tex]-17=a*-5^{2}+b*-5+c\\c=-25a+5b-17[/tex]

[tex]-17=a*3^{2}+b*3+c\\ c=-9a-3b-17[/tex]

We can solve these system of equations by substitution

Substitute [tex]c=-9a-3b-17[/tex]

[tex]-9a-3b-17=25a+5b-17\\-9a-3b-17=-2[/tex]

Isolate a for the first equation

[tex]-9a-3b-17=-25a+5b-17\\a=\frac{b}{2}[/tex]

Substitute [tex]a=\frac{b}{2}[/tex] into the second equation

[tex]-9\left(-\frac{b}{2}\right)-3b-17=-2[/tex]

Find the value of b

[tex]-9\left(-\frac{4b}{17}\right)-3b-17=-2\\ b=-2[/tex]

Find the value of a

[tex]a=\frac{b}{2}\\ a=-1[/tex]

The solutions to the system of equations are:

b=-2,a=-1,c=-2

So the quadratic function whose graph contains these points is

[tex]y=-x^{2}-2x-2[/tex]

As you can corroborate with the graph of this function.