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.