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Let's go over the questions in the image you shared. Here's a step-by-step guide to solving the problems related to rates of change.

### 1. **Finding the rate of change from two points (Questions 1 and 2):**
   The rate of change (or slope) between two points on a line is given by:

   \[
   \text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1}
   \]

#### **Question 1**:
   The table provides two sets of points: (5, 2), (10, 3), (15, 4), and (20, 5). To find the rate of change using two points, let's use (5, 2) and (10, 3).
   \[
   \frac{3 - 2}{10 - 5} = \frac{1}{5}
   \]
   So, the rate of change is \( \frac{1}{5} \).

#### **Question 2**:
   The table provides two points: (1, 15) and (4, -3). Using these points:
   \[
   \frac{-3 - 15}{4 - 1} = \frac{-18}{3} = -6
   \]
   The rate of change is -6.

---

### 2. **Population Density (Question 3):**
   The table provides population density data over several decades.

   You are asked to find the average rate of change between 2000 and 2009. From the table:
   - In 2000, the population density was 79.6.
   - In 2009, the population density was 96.7.

   Using the rate of change formula:
   \[
   \frac{96.7 - 79.6}{2009 - 2000} = \frac{17.1}{9} \approx 1.9 \text{ people per square mile per year}.
   \]

---

### 3. **Band Population (Question 4):**
   In 2012, there were 275 students in the band, and in 2018, there were 305 students. The question asks for the annual rate of change between 2012 and 2018.
   \[
   \frac{305 - 275}{2018 - 2012} = \frac{30}{6} = 5 \text{ students per year}.
   \]

---

### 5. **Temperature (Question 5):**
   The graph shows temperatures at different times during a day.

#### a. **6 A.M. to 7 A.M.:**
   From the graph:
   - At 6 A.M., the temperature is about 60°F.
   - At 7 A.M., the temperature is about 70°F.

   Using the rate of change formula:
   \[
   \frac{70 - 60}{7 - 6} = \frac{10}{1} = 10 \text{ degrees per hour}.
   \]
   This means the temperature increased by 10°F per hour between 6 A.M. and 7 A.M.

#### b. **1 P.M. to 2 P.M.:**
   - At 1 P.M., the temperature is about 90°F.
   - At 2 P.M., the temperature is about 95°F.

   Using the rate of change formula:
   \[
   \frac{95 - 90}{2 - 1} = \frac{5}{1} = 5 \text{ degrees per hour}.
   \]
   This means the temperature increased by 5°F per hour between 1 P.M. and 2 P.M.

---

### 6. **Coal Exports (Question 6):**

#### a. **2000 to 2002:**
   From the graph:
   - In 2000, coal exports were about 50 million short tons.
   - In 2002, coal exports were about 60 million short tons.

   Using the rate of change formula:
   \[
   \frac{60 - 50}{2002 - 2000} = \frac{10}{2} = 5 \text{ million short tons per year}.
   \]

#### b. **2005 to 2006:**
   - In 2005, coal exports were about 60 million short tons.
   - In 2006, coal exports were about 55 million short tons.

   Using the rate of change formula:
   \[
   \frac{55 - 60}{2006 - 2005} = \frac{-5}{1} = -5 \text{ million short tons per year}.
   \]
   The negative value means coal exports decreased by 5 million short tons per year.

---

Would you like further explanations or details on any part of these solutions?

Here are 5 related questions to deepen your understanding:
1. How do you interpret a negative rate of change in real-world scenarios?
2. What does the rate of change tell you about the trend in population density over time?
3. How can the rate of change formula be used to predict future values?
4. In what situations would you expect the rate of change to be zero?
5. How can you compare rates of change between different contexts (e.g., temperature vs. population)?

**Tip**: Always use points that are easy to extract from the graph or table when calculating the rate of change for more accurate results.

Find the rate of change of the function using points from tables, population density changes, temperature changes, and coal export changes over time.

Learn how to calculate the rate of change in various contexts, including population density, temperature over time, and coal exports. This lesson covers key concepts such as slope and rate of change, using real-world data for students in Grades 8-10.

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