https://file.aiquickdraw.com/mathbot/file/1726768541394yfikiub5.jpg

Let's analyze the information from the image and solve the questions step by step.

1. **Equation Provided (Function 2):**
   \[
   f(x) = -3x^2 - 12x - 8
   \]
   This is a quadratic function, and the general form of a quadratic equation is:
   \[
   f(x) = ax^2 + bx + c
   \]
   In this case:
   - \( a = -3 \)
   - \( b = -12 \)
   - \( c = -8 \)

2. **Finding the vertex of Function 2:**
   The vertex \( (h, k) \) of a quadratic function can be found using the formula:
   \[
   h = \frac{-b}{2a}, \quad k = f(h)
   \]
   Let's calculate the vertex of this function.

   \[
   h = \frac{-(-12)}{2(-3)} = \frac{12}{-6} = -2
   \]
   Now, substitute \( h = -2 \) into the equation to find \( k \):
   \[
   k = -3(-2)^2 - 12(-2) - 8 = -3(4) + 24 - 8 = -12 + 24 - 8 = 4
   \]
   Therefore, the vertex of **Function 2** is \( (-2, 4) \).

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Next, let's analyze **Function 1** by interpreting the graph provided.

3. **Vertex of Function 1:**
   From the graph of Function 1, it appears that the vertex is at the point \( (1, 5) \). This can be determined visually as the highest point on the parabola.

---

Now, let's answer the specific questions from the image:

### (a) What is the vertex of Function 1?
   The vertex of Function 1 is \( (1, 5) \).

### (b) What is the vertex of Function 2?
   The vertex of Function 2 is \( (-2, 4) \).

### (c) Which function has the larger maximum value?
   The maximum value is given by the \( y \)-coordinate of the vertex for both functions:
   - Maximum value of Function 1: 5
   - Maximum value of Function 2: 4

   Therefore, **Function 1** has the larger maximum value.

---

Would you like more details on any of these steps, or explanations on specific concepts?

Here are some follow-up questions you might find helpful:

1. How is the vertex form of a quadratic function derived from its standard form?
2. Can you explain how the sign of \( a \) (the coefficient of \( x^2 \)) affects the shape of the parabola?
3. What are the steps to convert a quadratic function from standard form to vertex form?
4. How do you find the axis of symmetry of a parabola?
5. How can the discriminant of a quadratic equation help determine the nature of its graph?

**Tip:** Always double-check your calculations when finding the vertex, as small errors in substitution can lead to incorrect results!

Analyze the graph and quadratic equation to determine the vertex of each function and compare their maximum values.

Explore how to find the vertex of two quadratic functions and compare their maximum values. This problem focuses on interpreting a quadratic equation and a graph to determine the highest point of each function. Learn how to calculate the vertex and identify which function has the larger maximum value.

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