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The given problem is asking for the equation of a parabola that passes through the point (-7, 15) and has its vertex at (-5, 3).

A parabola's vertex form is written as:

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

where \((h, k)\) is the vertex, and \(a\) is a constant that determines the width and direction of the parabola.

### Step 1: Identify the vertex and substitute into the vertex form.
From the problem, the vertex is \((-5, 3)\), so:
\[
y = a(x + 5)^2 + 3
\]

### Step 2: Use the point (-7, 15) to find the value of \(a\).
Substitute \((-7, 15)\) into the equation, where \(x = -7\) and \(y = 15\):

\[
15 = a(-7 + 5)^2 + 3
\]
\[
15 = a(-2)^2 + 3
\]
\[
15 = 4a + 3
\]
\[
12 = 4a
\]
\[
a = 3
\]

### Step 3: Write the equation of the parabola.
Now that we have \(a = 3\), the equation becomes:

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

### Conclusion:
The correct equation of the parabola is option **2) \( y = 3(x + 5)^2 + 3 \)**.

Let me know if you want further details or explanations!

Here are 5 related questions:
1. How do you find the vertex form of a parabola given a graph?
2. What is the difference between the vertex form and standard form of a quadratic equation?
3. How do you determine whether a parabola opens upwards or downwards?
4. How do you convert the vertex form of a parabola to its standard form?
5. How can you determine the focus and directrix of a parabola from its equation?

**Tip**: In the vertex form of a parabola, the sign of the coefficient \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

What is the equation of a parabola that passes through (-7, 15) and has its vertex at (-5, 3)?

Find the equation of a parabola that passes through the point (-7, 15) and has a vertex at (-5, 3). This problem explores key algebraic concepts like vertex form of a quadratic equation. Ideal for students in Grades 9-11.

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