1. What is an Exponential Function?

An exponential function is a mathematical function of the form y = a e^{b x}, where e is Euler's number (approximately 2.71828). These functions model growth or decay processes where the rate of change is proportional to the current value.

2. How Does the Calculator Work?

The calculator plots the exponential function:

Where:

• \( y \) — Dependent variable (unitless)
• \( x \) — Independent variable (unitless)
• \( a \) — Initial value coefficient (unitless)
• \( b \) — Growth/decay rate coefficient (unitless)
• \( e \) — Euler's number (≈2.71828)

Explanation: When b > 0, the function shows exponential growth. When b < 0, it shows exponential decay. The value of a determines the y-intercept.

3. Importance of Exponential Functions

Details: Exponential functions are fundamental in modeling population growth, radioactive decay, compound interest, and many natural phenomena. They are essential in mathematics, physics, biology, and economics.

4. Using the Calculator

Tips: Enter the coefficients a and b, and the x-range you want to plot. The calculator will generate the exponential curve. Larger b values create steeper curves.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between exponential and linear growth?
A: Linear growth adds a fixed amount per time period, while exponential growth multiplies by a fixed factor.

Q2: How does changing 'a' affect the graph?
A: 'a' determines the y-intercept (value when x=0). It scales the entire graph vertically.

Q3: How does changing 'b' affect the graph?
A: 'b' determines the growth rate. Positive b gives growth, negative b gives decay. Larger absolute values make the curve steeper.

Q4: What are some real-world examples?
A: Population growth (b>0), radioactive decay (b<0), compound interest, and cooling/heating processes.

Q5: What's special about e in this equation?
A: The natural base e makes calculus operations simpler and appears naturally in many growth processes.