Parabola Equation

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Let's say you are given a task of finding the equation of a parabola passing through the tree given points (-2, 10), (1, -11) and (3, -5) in Cartesian coordinates, where the first number of the three coordinate-pairs is the x-coordinate. The second number of the three coordinates-pairs is the y-coordinate.

All equations describing a parabola can be written in the following form: ($bx$ means b times x) $$y=a{x}^2+bx+c$$

Now substitute the three given (x, y)-coordinates in the general form of a parabolic equation (above). Thus creating a system of three equations. The point (-2, 10) yields: (I will call this our first equation below) $$\mathrm{10}=a{\mathrm{(-2)}}^2+b\mathrm{(-2)}+c$$ $$\mathrm{10}=4a-2b+c$$

The point (1, -11) yields: (I will call this our second equation below) $$\mathrm{-11}=a{\mathrm{(1)}}^2+b\mathrm{(1)}+c$$ $$\mathrm{-11}=a+b+c$$

The point (3, -5) yields: (I will call this our third equation below) $$\mathrm{-5}=a{\mathrm{(3)}}^2+b\mathrm{(3)}+c$$ $$\mathrm{-5}=9a+3b+c$$

If we have three unknown variables and three equations, we can solve the problem. The second equation tells us that: $$\mathrm{-11}=a+b+c$$

In other words: $$c=-a-b-\mathrm{11}$$

The first equation tells us that: $$\mathrm{10}=4a-2b+c$$

Now substitute $c$ in the equation above for $(-a-b-\mathrm{11})$

This yields: $$\mathrm{10}=4a-2b+(-a-b-\mathrm{11})$$ Isolating $b$ yields: $$\mathrm{21}=3a-3b$$ $$b=\frac{3a-21}{3}$$

The third equation tells us that: $$\mathrm{-5}=9a+3b+c$$

Now substitute $b$ and $c$ for the values we have found via isolation above. Equation number three becomes:

$$\mathrm{-5}=9a+3\left(\frac{3a-21}{3}\right)+(-a-b-\mathrm{11})$$ $c$ contained a $b$. Replacing it yields: (now we only have one unknown, which is $a$) $$\mathrm{-5}=9a+3\left(\frac{3a-21}{3}\right)+(-a-(\frac{3a-21}{3})-\mathrm{11})$$ Let's try to isolate $a$. $$\mathrm{-5}=10a-\mathrm{21}+7-11$$ $$\mathrm{-5}=10a-\mathrm{25}$$ $$\mathrm{-1}=2a-5$$ $$a=2$$ Now we use our result for $a$ in order to find a numerical value for the coefficient $b$. Remember that $b$ was: $$b=\frac{3a-21}{3}$$ $$b=\frac{3\mathrm{(2)}-21}{3}$$ $$b=-5$$ Now we use our result for $b$ in order to find a numerical value for the coefficient $c$. Remember that $c$ was: $$c=-a-b-\mathrm{11}$$ $$c=-\mathrm{(2)}-(-\mathrm{5)}-\mathrm{11}$$ $$c=-8$$

And our final version of the equation for the parabola passing through the three given points (-2, 10), (1, -11) and (3, -5) in Cartesian coordinates becomes: $$y=2x^2-5x-8$$