What Are Exponential Functions? 🚀
**Exponential functions** describe situations where a quantity grows or shrinks at a constant rate over time. They are in the form $$y = ab^x$$, where:
- **a** is the **initial value** (the value of $$y$$when$$x=0$$).
- **b** is the **growth or decay factor**. If $$b > 1$$, it's growth. If $$0 < b < 1$$, it's decay.
Unlike linear functions that have a constant slope, exponential functions have a constant ratio. The graph is a smooth curve that gets very steep very quickly! [Image of a graph of an exponential function]
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Video: Graphing an Exponential Function 🎬
This video will show you how to graph an exponential function using the initial value and growth factor.
Practice Problems
Solve the following problems and check your answers.
1) What is the initial value?
$$y = 5 \cdot (2)^x$$
$$y = 5 \cdot (2)^x$$
Initial value (a):
2) What is the growth factor?
$$y = 20 \cdot (1.5)^x$$
$$y = 20 \cdot (1.5)^x$$
Growth factor (b):
3) Evaluate the function at $$x=2$$:
$$y = 3 \cdot (4)^x$$
$$y = 3 \cdot (4)^x$$
Value:
4) A population of bacteria starts at 100 and doubles every hour. How many bacteria are there after 3 hours?
Bacteria:
⚡ Exponential Speed Quiz
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