Hello guys, please help me, knowing that the parabola passes through the points A(0,1), B(-1,-2) e C(-2,7). How can i find the equation?
The equation for a parabola would be knowing the values of $\displaystyle a, b, c$ in $\displaystyle y = ax^2 + bx + c$.
What does C = (-2,7) on the parabola tell you? It says that when x = -2, the y value of that formula will be 7.
Thus $\displaystyle (7) = a(-2)^2 + b(-2) + c = 4a - 2b + c$.
So C on the parabola tells you that $\displaystyle 4a - 2b + c = 7$.
A and B on the parabola will likewise tell you something. And if you know what a, b, and c are, then you have "found the parabola".
Let:
$\displaystyle y(x)=ax^2+bx+c$
From the given points, this gives the 3X3 linear system:
$\displaystyle a(0)^2+b(0)+c=1$
$\displaystyle a(-1)^2+b(-1)+c=-2$
$\displaystyle a(-2)^2+b(-2)+c=7$
Simplified:
$\displaystyle c=1$ this allows us to reduce the problem to the following 2X2 system
$\displaystyle a-b=-3$
$\displaystyle 4a-2b=6$
Solve this system, and you will have the values of the coefficients $\displaystyle a,b,c$ which will give you the parabola you want.