Algebra worksheet focusing on calculating and interpreting the rate of change using tables and graphs.

It looks like you've uploaded a worksheet titled "Algebra: Slope and Rate of Change" with two tables and one graph. Let's go through each part step by step to solve the problems and explain what the rate of change means in each situation.

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Problem 1: Table – Time vs. Cost

| Time (hours) | Cost ($) |
|--------------|----------|
| 0 | 50 |
| 1 | 80 |
| 2 | 110 |
| 3 | 140 |

We are told that the rate of change is constant, so we can calculate it using any two points.

#### Step 1: Use the formula for rate of change:
$$
\text{Rate of change} = \frac{\text{Change in Cost}}{\text{Change in Time}} = \frac{\Delta y}{\Delta x}
$$

Let’s use the first two points: (0, 50) and (1, 80)

$$
\frac{80 - 50}{1 - 0} = \frac{30}{1} = 30
$$

Check with another pair: (1, 80) and (2, 110)

$$
\frac{110 - 80}{2 - 1} = \frac{30}{1} = 30
$$

So, the rate of change is $30 per hour.

#### What does this mean?
The cost increases by $30 for every additional hour. This could represent a service charge such as a repair technician who charges a base fee of $50 plus $30 per hour.

> ✔ Answer: The rate of change is $30 per hour. It means the cost increases by $30 for each additional hour of service.

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Problem 2: Table – Time vs. Distance

| Time (seconds) | Distance (feet) |
|----------------|-----------------|
| 0 | 0 |
| 4 | 16 |
| 8 | 32 |
| 10 | 40 |

Again, rate of change is constant.

Use points (0, 0) and (4, 16):

$$
\frac{16 - 0}{4 - 0} = \frac{16}{4} = 4
$$

Check with (4, 16) and (8, 32):

$$
\frac{32 - 16}{8 - 4} = \frac{16}{4} = 4
$$

So, the rate of change is 4 feet per second.

#### What does this mean?
The object is moving at a constant speed of 4 feet per second. This is the speed of the object.

> ✔ Answer: The rate of change is 4 feet per second. It represents the speed of the object — it travels 4 feet every second.

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Problem 3: Graph – Find the rate of change

We have a graph showing a straight line from (0, 0) to (4, 8). The x-axis is labeled "Time (min)", and the y-axis is labeled "Distance (miles)".

We need to find the slope of the line, which is the rate of change.

#### Step 1: Pick two points on the line.
From the graph:
- At time = 0 min, distance = 0 miles → (0, 0)
- At time = 4 min, distance = 8 miles → (4, 8)

$$
\text{Rate of change} = \frac{8 - 0}{4 - 0} = \frac{8}{4} = 2
$$

So, the rate of change is 2 miles per minute.

But let's interpret it meaningfully.

Wait — 2 miles per minute is extremely fast (that's 120 mph!), but perhaps it's a typo or the units are different?

Let’s double-check the graph:

- The x-axis goes from 0 to 4 (labeled "Time (min)")
- The y-axis goes from 0 to 8 (labeled "Distance (miles)")

Yes, so 8 miles in 4 minutes → 2 miles per minute.

But that’s not realistic unless it's a car or something very fast.

Alternatively, maybe it's 2 miles per minute? Or perhaps the graph is just an example.

But mathematically, based on the graph:

$$
\text{Rate of change} = \frac{8 \text{ miles}}{4 \text{ min}} = 2 \text{ miles per minute}
$$

But let's convert it to more standard units:
2 miles/min × 60 min/hour = 120 miles per hour.

That’s fast, but possible for a vehicle.

#### What does this mean?
The object is moving at a constant speed of 2 miles per minute (or 120 mph), meaning it covers 2 miles every minute.

> ✔ Answer: The rate of change is 2 miles per minute. It represents the speed of the object — it travels 2 miles for every minute that passes.

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✔ Final Answers Summary:

1. Table 1 (Cost vs. Time):
- Rate of change: $30 per hour
- Meaning: The cost increases by $30 for each additional hour of service.

2. Table 2 (Distance vs. Time):
- Rate of change: 4 feet per second
- Meaning: The object moves at a constant speed of 4 feet per second.

3. Graph (Distance vs. Time):
- Rate of change: 2 miles per minute
- Meaning: The object travels 2 miles every minute (very fast, but mathematically correct).

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Let me know if you'd like help converting units or interpreting the graph differently!