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To find the equation of a parabola given its vertex and a point on the parabola, we can use the vertex form of a parabola's equation, which is:

\[ y = a(x - h)^2 + k \]

where \((h, k)\) is the vertex of the parabola. For the given problem:

Vertex: \((-1, 5)\)
Point: \((-3, -7)\)

First, we substitute the vertex \((h, k)\) into the vertex form equation:

\[ y = a(x + 1)^2 + 5 \]

Next, we use the given point \((-3, -7)\) to find the value of \(a\). We substitute \((-3, -7)\) into the equation:

\[ -7 = a(-3 + 1)^2 + 5 \]
\[ -7 = a(-2)^2 + 5 \]
\[ -7 = 4a + 5 \]

Now, solve for \(a\):

\[ -7 - 5 = 4a \]
\[ -12 = 4a \]
\[ a = -3 \]

Thus, the equation of the parabola is:

\[ y = -3(x + 1)^2 + 5 \]

Do you want any further details or have any questions? Here are some related questions to consider:

1. How can you verify the equation of a parabola given different points?
2. What is the standard form of a parabolic equation?
3. How do you convert the vertex form of a parabola to the standard form?
4. What are the properties of parabolas that can be derived from their equations?
5. How can you determine the direction of a parabola's opening from its equation?

**Tip:** When solving for the equation of a parabola, always ensure to correctly substitute and simplify step by step to avoid errors in finding the value of \(a\).

Find the equation of a parabola given the vertex (-1, 5) and the point (-3, -7).

Learn how to find the equation of a parabola using its vertex (-1, 5) and a point (-3, -7) with a detailed step-by-step solution. This problem covers key algebraic concepts, including the vertex form of a parabola. Suitable for students in Grades 9-11.

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