# Finding the equation of the parabola

#### Chipset3600

##### Member
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?

#### MarkFL

Staff member
This question has also been posted on MHF for which responses have been given.

I don't want to see the folks here take the time to post help when it has already been given elsewhere. #### soroban

##### Well-known member
Hello, Chipset3600!

Find the equation of the parabola passing through: A(0,1), B(-1,-2), C(-2,7).

There are two such parabolas: one "vertical" $$\cup$$, the other "horizontal" $$\supset$$.

Vertical: .$$y \:=\:ax^2 + bx + c$$

Substitute the points and create a system of three equations.
The system has the solution: .$$a = 6,\:b = 9,\:c = 1$$

The equation is: .$$y \;=\;6x^2 + 9x + 1$$

Horizontal: .$$x \;=\;ay^2 + by + c$$

Substitute the points and create a system of three equations.
The system has the solution: .$$a = \text{-}\tfrac{2}{27},\:b = \tfrac{7}{27},\:c = \text{-}\tfrac{5}{27}$$

The equation is: .$$x \;=\;\text{-}\tfrac{2}{27}y^2 + \tfrac{7}{27}y - \tfrac{5}{27}$$

#### MarkFL

I didn't consider anything but the parabola with vertical axis of symmetry...I suppose we could find an infinite number of parabolas by rotating the axes by any angle we choose. Thanks guys 