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Let's go through the first two questions from the image you provided.

### Question 1:
**State whether or not the following relations are functions. Include the reason for your choice.**

#### Part a)
For part **a)**, a graph is provided. To determine if this relation is a function, we can use the **vertical line test**: if any vertical line crosses the graph at more than one point, it is **not** a function. 

Looking at the graph, it seems that every vertical line drawn would only cross the curve at one point, indicating that the graph **is** a function.

#### Part b)
The relation is given as a set of ordered pairs: \(\{(2, 4), (2, 6), (2, 8), (2, 10), (2, 12)\}\).

For a relation to be a function, each **input** (first value in the pair) must correspond to exactly **one output**. In this case, the input "2" maps to multiple outputs (4, 6, 8, 10, 12). This violates the definition of a function, so this relation **is not a function**.

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### Question 2:
The path of a thrown baseball can be modeled by the quadratic function:
\[
h(d) = -0.005d^2 + 0.25d + 2
\]
where \(h\) is the height of the ball in meters, and \(d\) is the horizontal distance of the ball from the player in meters.

#### Part a)
**What is the maximum height reached by the ball?**

Since the equation is quadratic and has a negative coefficient for \(d^2\), the parabola opens downward, and the maximum height is at the vertex of the parabola.

To find the vertex, use the formula for the horizontal coordinate of the vertex:
\[
d = -\frac{b}{2a}
\]
where \(a = -0.005\) and \(b = 0.25\). Substituting these values in:
\[
d = -\frac{0.25}{2(-0.005)} = 25
\]
Now that we have \(d = 25\), substitute this back into the equation for \(h(d)\) to find the maximum height:
\[
h(25) = -0.005(25)^2 + 0.25(25) + 2
\]
Let's compute this.

#### Part b)
**What is the horizontal distance of the ball from the player when the ball reaches its maximum height?**

This has already been found as \(d = 25\) meters, which is the horizontal distance at the vertex.

I'll calculate the height from part a.The horizontal distance at which the ball reaches its maximum height is **25 meters**, and the maximum height reached by the ball is **5.125 meters**.

---

### Do you need any further clarifications or details on these problems?

Here are 5 related questions that expand on this topic:
1. How can you identify whether a relation is a function using a set of points?
2. What are the characteristics of a quadratic function that opens downward versus upward?
3. How can you determine the maximum or minimum point of any quadratic function?
4. Can you explain why having multiple outputs for the same input disqualifies a relation from being a function?
5. How would changing the coefficients in the quadratic equation affect the path of the baseball?

**Tip:** For any quadratic function \( ax^2 + bx + c \), the vertex (maximum or minimum) is always at \( x = -\frac{b}{2a} \), and this helps find optimal values such as maximum height or minimum cost in real-life applications.

State whether or not the following relations are functions and solve the quadratic equation given for the path of a thrown baseball.

Explore whether certain relations qualify as functions and how to model the path of a baseball using quadratic equations. This guide covers the function definition, uses of the vertical line test, and solving for the vertex of a parabola to determine maximum height.

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