what is the formula for a parbela passing through (-2,0), (3,0) and (0,6)

To find the equation of a parabola that passes through the points \((-2, 0)\), \((3, 0)\), and \((0, 6)\), we can use the general form of a quadratic equation:

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

Since the points \((-2, 0)\) and \((3, 0)\) are the **roots** of the parabola, we know the equation can be written in **factored form** as:

\[
y = a(x + 2)(x - 3)
\]

Next, we will use the point \((0, 6)\) to determine the value of \(a\). Substituting \(x = 0\) and \(y = 6\) into the factored form:

\[
6 = a(0 + 2)(0 - 3)
\]

\[
6 = a(2)(-3)
\]

\[
6 = -6a
\]

Solving for \(a\):

\[
a = -1
\]

Thus, the equation of the parabola is:

\[
y = -(x + 2)(x - 3)
\]

We can expand this to get the standard form:

\[
y = -(x^2 - x - 6)
\]

\[
y = -x^2 + x + 6
\]

Therefore, the equation of the parabola is:

\[
y = -x^2 + x + 6
\]

Would you like a detailed explanation of the expansion, or have any further questions?

Here are some related questions:
1. How do you convert between standard form and factored form of a quadratic equation?
2. How do you find the vertex of a parabola given its equation?
3. How do you calculate the axis of symmetry for a parabola?
4. What is the geometric interpretation of the coefficient \(a\) in a parabola's equation?
5. How do you determine the direction of opening for a parabola?

**Tip:** The sign of the coefficient \(a\) in a parabola’s equation tells you whether it opens upward (positive \(a\)) or downward (negative \(a\)).

What is the formula for a parabola passing through (-2,0), (3,0), and (0,6)?

Learn how to find the equation of a parabola that passes through the points (-2,0), (3,0), and (0,6) using algebraic concepts like factored form and standard form of quadratic equations. This detailed solution is ideal for students in Grades 9-11.

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