Find the equation of the parabola passing through (0,3),(1,0) and (3,0)

To find the equation of the parabola passing through the points \((0, 3)\), \((1, 0)\), and \((3, 0)\), we can assume the general form of a quadratic equation:

\[
y = ax^2 + bx + c
\]

Given the points:

1. \((0, 3)\) gives us the equation: 
   \[
   3 = a(0)^2 + b(0) + c \implies c = 3
   \]

2. \((1, 0)\) gives us the equation:
   \[
   0 = a(1)^2 + b(1) + 3 \implies a + b + 3 = 0 \implies a + b = -3
   \]

3. \((3, 0)\) gives us the equation:
   \[
   0 = a(3)^2 + b(3) + 3 \implies 9a + 3b + 3 = 0 \implies 3a + b = -1
   \]

Now, we have two equations:
\[
\begin{aligned}
1.\quad & a + b = -3 \\
2.\quad & 3a + b = -1
\end{aligned}
\]

To solve for \(a\) and \(b\), subtract the first equation from the second:

\[
(3a + b) - (a + b) = -1 - (-3)
\]

Simplifying, we get:
\[
2a = 2 \implies a = 1
\]

Substitute \(a = 1\) back into the first equation:
\[
1 + b = -3 \implies b = -4
\]

So, the equation of the parabola is:
\[
y = x^2 - 4x + 3
\]

This is the required equation.

Would you like more details or have any questions? Here are some related questions:

1. How do we find the vertex of the parabola \(y = x^2 - 4x + 3\)?
2. How can we determine if a parabola opens upwards or downwards?
3. How do you find the focus and directrix of the parabola \(y = x^2 - 4x + 3\)?
4. What is the axis of symmetry of the parabola \(y = x^2 - 4x + 3\)?
5. How would the equation change if the parabola passed through a different set of points?

**Tip:** For any quadratic equation \(y = ax^2 + bx + c\), the vertex form can be helpful: \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

Learn how to find the equation of a parabola passing through points (0, 3), (1, 0), and (3, 0) step-by-step. This problem involves solving a system of equations and applying the quadratic formula. Suitable for high school students studying quadratic equations.

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