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 $(-2, 4)$.

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!