https://file.aiquickdraw.com/mathbot/file/1722008212917ax5359w5.PNG

Let's match the models to the equations given:

1. **Linear growth**: This is represented by an equation of the form \( y = mx + b \) where the slope \( m \) is positive.
    - Equation: \( y = 0.75x - 80 \)

2. **Linear decay**: This is represented by an equation of the form \( y = mx + b \) where the slope \( m \) is negative.
    - Equation: \( y = -3.2x + 200 \)

3. **Exponential growth**: This is represented by an equation of the form \( y = ae^{kt} \) where \( k \) is positive.
    - Equation: \( y = 400e^{0.078t} \)

4. **Exponential decay**: This is represented by an equation of the form \( y = ae^{kt} \) where \( k \) is negative.
    - Equation: \( y = 8000e^{-0.124t} \)

5. **Logistic growth**: This is represented by an equation of the form \( y = \frac{L}{1 + Ce^{-kt}} \).
    - Equation: \( y = \frac{12000}{1 + 79e^{-0.24t}} \)

So, the matched models are:

1. Linear growth: \( y = 0.75x - 80 \)
2. Linear decay: \( y = -3.2x + 200 \)
3. Exponential growth: \( y = 400e^{0.078t} \)
4. Exponential decay: \( y = 8000e^{-0.124t} \)
5. Logistic growth: \( y = \frac{12000}{1 + 79e^{-0.24t}} \)

Would you like more details or have any questions? Here are 5 questions you might ask next:
1. Can you explain the properties of exponential growth?
2. What are some real-life examples of logistic growth?
3. How do you identify linear equations from their graphs?
4. Can you show how to derive the logistic growth equation?
5. What are the differences between exponential growth and decay?

**Tip:** Always look at the exponent's sign in exponential functions to distinguish between growth and decay.

Explore different mathematical models such as linear growth, linear decay, exponential growth, exponential decay, and logistic growth. Learn their equations and applications with detailed examples and explanations.

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