# Parabola - equation from three points

Question: Find the equation of the parabola whose axis is parallel to the y-axis and which passes through the points (0,4) (1,9) and (-2,6)

Well as the parabola has its axis parallel to the y-axis and it is obviously opening upwards, the general form can be written as: $$(x-h)^2 = 4a(y-k)$$

Then I would have to go through the painstaking task of solving three variables from three equations. Obvious method to do it. I was wondering if there's any easier way of doing it?

## 1 Answer

The general formula for a parabola is $f(x)=ax^2+bx+c$.

• Plugging in $(0,4)$, we get: $c=4$.
• Plugging in $(1,9)$, we get: $a+b+c=9$.
• Plugging in $(-2,6)$, we get: $4a-2b+c=6$.

The system of equations to be solved becomes:

$\begin{cases}a+b=5 \\ 4a-2b=2 \end{cases}$

This gives $a=2$ and $b=3$.

The formula for your parabola becomes: $f(x)=2x^2+3x+4$.

• I see. But any easier way to do it? – Gummy bears Sep 10 '14 at 18:59
• Even easier? No, I don't think so. – rae306 Sep 10 '14 at 19:00
• What is easier than posting on Mathematics and having someone else solve it for you? – ja72 Sep 10 '14 at 19:02
• @ja72 The point isn't to get the answer. Rather to get a method to use in future problems......... – Gummy bears Sep 10 '14 at 19:03
• 48 seconds left lol – Gummy bears Sep 10 '14 at 19:05