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To find the vertex and axis of symmetry of the quadratic function \( f(x) = 3(x + 5)^2 + 1 \), let's follow these steps:

### 1. Vertex Form
The function is given in the vertex form of a quadratic equation:
\[
f(x) = a(x-h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.

### 2. Identify the Vertex
From the given function \( f(x) = 3(x + 5)^2 + 1 \), we can see that:
- \( h = -5 \) (because \(x + 5 = x - (-5)\))
- \( k = 1 \)

So, the vertex \((h, k)\) is \((-5, 1)\).

### 3. Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex, and its equation is:
\[
x = h
\]
Thus, the axis of symmetry is \( x = -5 \).

### Final Answers:
- The vertex is \((-5, 1)\).
- The axis of symmetry is \( x = -5 \).

Would you like further details or have any questions?

Here are some related questions you might find useful:

1. What does the coefficient of the squared term (in this case, 3) tell us about the parabola?
2. How does the value of \( h \) affect the horizontal shift of the parabola?
3. How does the value of \( k \) affect the vertical shift of the parabola?
4. What are the steps to convert a standard form quadratic function into vertex form?
5. How can you determine if a parabola opens upwards or downwards?

**Tip:** The vertex form of a quadratic equation makes it easy to identify the vertex and axis of symmetry, which is useful in graphing and understanding the behavior of the function.

Find the vertex and axis of symmetry of f(x) = 3(x + 5)^2 + 1.

Learn how to find the vertex and axis of symmetry of the quadratic function f(x) = 3(x + 5)^2 + 1 using the vertex form equation. This problem covers key algebraic concepts, suitable for students in Grades 9-12.

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