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Let's solve the quadratic function \( y = 3x^2 + 18x + 24 \) by completing the square to find the vertex and determine whether it is a maximum or minimum.

### Step 1: Factor out the coefficient of \( x^2 \)
First, factor out the \( 3 \) from the first two terms:

\[
y = 3(x^2 + 6x) + 24
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( x \) inside the parentheses (which is 6), square it, and add and subtract it inside the parentheses:

\[
3(x^2 + 6x + 9 - 9) + 24
\]

This simplifies to:

\[
3((x + 3)^2 - 9) + 24
\]

### Step 3: Simplify the expression
Now distribute the \( 3 \) and combine like terms:

\[
3(x + 3)^2 - 27 + 24
\]

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

### Step 4: Identify the vertex
The equation is now in vertex form \( y = a(x - h)^2 + k \), where the vertex is at \( (h, k) \). In this case, the vertex is:

\[
(-3, -3)
\]

### Step 5: Determine if the vertex is a maximum or minimum
Since the coefficient of \( (x + 3)^2 \) is positive (3), the parabola opens upwards, meaning the vertex is a **minimum**.

### Final Answer:
- **Vertex**: \( (-3, -3) \)
- The vertex represents a **minimum**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you solve a quadratic equation by completing the square?
2. What is the significance of the vertex in a parabola?
3. How can you tell if a parabola has a maximum or minimum point?
4. What is the general form of a quadratic equation in vertex form?
5. How does the sign of the leading coefficient affect the direction of a parabola?

**Tip**: The vertex form of a quadratic equation is a powerful way to easily identify the vertex and whether the function has a maximum or minimum point.

Complete the square for the quadratic function y = 3x^2 + 18x + 24 and determine the following: a) Identify the vertex as a maximum or minimum, b) Identify the coordinates of the vertex.

Learn how to complete the square for the quadratic function y = 3x^2 + 18x + 24 to find the vertex. This detailed solution shows how to rewrite the equation in vertex form and determine whether the vertex represents a maximum or minimum.

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