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# System prepended metadata

title: Linear vs. Exponential Functions Lesson

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# **TOPIC 2.2: Change in Linear vs. Exponential Functions**

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### **Part 1: Key Concepts - Take Notes! (8 min)**

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#### **Linear Functions (Constant Change)**
- **Pattern**: Same amount ADDED each time
- **Rate of Change**: Constant (slope/common difference)
- **Example**: $y = 2x + 3$
  - When x increases by $1$, y increases by $2$ (always)
- **Graph**: Straight line
- **Real-world**: Wages per hour, steady savings, constant speed

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#### **Exponential Functions (Proportional Change)**
- **Pattern**: Same amount MULTIPLIED each time
- **Rate of Change**: Proportional (percent increase/decrease)
- **Example**: $y = 3(2)^x$
  - When x increases by $1$, y is multiplied by $2$ (always)
- **Graph**: Curved (J-shape or decay curve)
- **Real-world**: Population growth, compound interest, viral spread, radioactive decay

#### **How to Identify from a Table:**

| $x$ | $y$ | 1st Differences | Ratios |
|---|---|---|---|
| $0$ | $3$ | - | - |
| $1$ | $7$ | + $4$ | × $2.33$|
| $2$ | $11$ | + $4$ | × $1.57$ |
| $3$ | $15$ | + $4$ | × $1.36$ |


**Linear** = constant first differences
**Exponential** = constant ratios

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### **Group Practice (17 min)**
Show all of your work in your notebooks.  Enter your answers here on the **[link](https://docs.google.com/forms/d/e/1FAIpQLSeOBEQ2BoB_fJYWY0YOCojv64gkAIoKY2Na-2H2j_HcaStFHw/viewform?usp=publish-editor)**.

**REMEMBERING:**
1. Define the difference between constant change and proportional change. Which type of function has a constant rate of change?

**UNDERSTANDING:**
2. Look at this table:
   
   | x | 0 | 1 | 2 | 3 |
   |---|---|---|---|---|
   | f(x) | 5 | 15 | 45 | 135 |
   
   Explain whether this is linear or exponential and describe how the y-values are changing.

**APPLYING:**
3. A phone plan charges $40 per month. A bank account doubles every year starting at $40.
   - Write an equation for each situation
   - Calculate the value after 6 months/years for each
   - Which is linear and which is exponential?

**ANALYZING:**
4. Examine these three tables. Classify each as linear, exponential, or neither. Show your work by calculating differences or ratios.

   **Table A:**
   | x | 1 | 2 | 3 | 4 |
   |---|---|---|---|---|
   | y | 8 | 11 | 14 | 17 |

   **Table B:**
   | x | 0 | 1 | 2 | 3 |
   |---|---|---|---|---|
   | y | 2 | 6 | 18 | 54 |

   **Table C:**
   | x | 1 | 2 | 3 | 4 |
   |---|---|---|---|---|
   | y | 3 | 6 | 11 | 18 |

**EVALUATING:**
5. A company claims their growth is "exponential" because they're "growing faster each year." Their revenue data:
   - Year 1: $100,000
   - Year 2: $150,000
   - Year 3: $200,000
   - Year 4: $250,000
   
   Evaluate their claim. Is this actually exponential growth? Justify your answer with mathematical evidence.

## **HOMEWORK:**
6. Design two different real-world scenarios—one that models linear growth and one that models exponential growth. For each:
   - Describe the scenario in 1-2 sentences
   - Create a table showing at least 4 data points
   - Write the explicit function
   - Sketch a quick graph showing the different growth patterns

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