Answer:
[tex]y=3x^{2} -x+5[/tex]
Step-by-step explanation:
The general form of a parabola is
[tex]y=ax^{2} +bx+c[/tex]
Now, we have three points, where each of them gives values for (x,y). We can use them to create a system of three equations with three unknown variables
[tex]5=c\\7=a(1)^{2} +b(1)+c\\19=a(-2)^{2} +b(-2)+c[/tex]
Then, we replace the value of [tex]c[/tex] in the second and third equation to find create an equation with only two variables
[tex]7=a+b+5\\19=4a-2b+5[/tex]
[tex]2=a+b\\14=4a-2b[/tex]
Then, we multiply the first equation by 2, and sum both equations
[tex]4=2a+2b\\14=4a-2b[/tex]
[tex]18=6a\\a=\frac{18}{6}\\ a=3[/tex]
Finally, we use this value to find [tex]b[/tex]
[tex]2=a+b\\2=3+b\\b=2-3\\b=-1[/tex]
Therefore, the equation of the parabola is
[tex]y=ax^{2} +bx+c\\y=3x^{2} -x+5[/tex]
