Understanding the characteristics of linear and exponential functions is crucial for identifying and analyzing these types of functions. Here’s a breakdown of their key characteristics:

Linear Functions

1. General Form: $N(t) = mt + b$

   • Where $m$ is the slope (rate of change), and $b$ is the y-intercept (initial value when $t = 0$).

2. Type:

   • Linear functions represent a constant rate of change.
   • The graph of a linear function is a straight line.

3. Behavior:

   • Increasing: If $m > 0$, the function increases as $t$ increases.
   • Decreasing: If $m < 0$, the function decreases as $t$ increases.
   • Constant: If $m = 0$, the function remains constant, meaning there’s no change as $t$ increases.

4. Rate of Change:

   • The rate of change is constant and equal to the slope $m$.
   • For each unit increase in $t$, $N(t)$ changes by $m$.

5. Example:

   • $N(t) = 5t + 3$: A linear function with a slope of 5 (increasing by 5 units per time unit) and an initial value of 3.

Exponential Functions

1. General Form: $N(t) = a \times r^t$

   • Where $a$ is the initial value (when $t = 0$), and $r$ is the base (growth or decay factor).

2. Type:

   • Exponential functions represent a rate of change that is proportional to the current value.
   • The graph of an exponential function is a curve that either rises or falls exponentially.

3. Behavior:

   • Exponential Growth: If $r > 1$, the function exhibits growth; it increases rapidly as $t$ increases.
   • Exponential Decay: If $0 < r < 1$, the function exhibits decay; it decreases rapidly as $t$ increases.

4. Rate of Change:

   • The rate of change is not constant; it increases or decreases at a rate proportional to the current value.
   • The function grows or decays by a consistent percentage per unit time.

5. Example:

   • $N(t) = 100 \times 1.05^t$: An exponential growth function with an initial value of 100 and a growth factor of 1.05 (5% increase per time unit).

Comparing Linear and Exponential Functions

• Linear functions have a constant rate of change, resulting in a straight-line graph.
• Exponential functions have a variable rate of change, leading to a curved graph that either steeply rises or falls, depending on whether it’s growth or decay.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How can you determine whether a real-world situation is best modeled by a linear or an exponential function?
2. What are some real-life examples where linear functions are used?
3. How does the slope of a linear function relate to the rate of change in practical terms?
4. What are the implications of exponential growth or decay in population modeling?
5. How would you graphically distinguish between linear and exponential functions on a coordinate plane?

Tip: When analyzing data, if the difference between successive values is constant, you’re likely dealing with a linear function. If the ratio of successive values is constant, it’s likely an exponential function.