1. What is an Exponential Function?

An exponential function is a mathematical function of the form y = a × b^x, where 'a' is a constant, 'b' is the base (positive and not equal to 1), and 'x' is the exponent. These functions model growth or decay processes.

2. How Does the Calculator Work?

The calculator plots the exponential function:

Where:

• \( a \) — Initial value or scaling factor (unitless)
• \( b \) — Base of the exponential (unitless, positive, ≠1)
• \( x \) — Independent variable (unitless)
• \( y \) — Dependent variable (unitless)

Explanation: The function shows rapid growth when b > 1 or decay when 0 < b < 1. The parameter 'a' scales the function vertically.

3. Applications of Exponential Functions

Details: Exponential functions model population growth, radioactive decay, compound interest, bacterial growth, and many natural phenomena.

4. Using the Calculator

Tips: Enter values for a and b, set the x-range for graphing. For decay functions, use b between 0 and 1. For growth, use b > 1.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between exponential and linear growth?
A: Exponential growth increases by multiplication (doubling, tripling), while linear growth adds fixed amounts.

Q2: Why can't the base 'b' be negative?
A: Negative bases lead to complex numbers when x is fractional, making real-world interpretation difficult.

Q3: What does it mean when b = 1?
A: The function becomes constant (y = a) since 1 to any power is 1.

Q4: How is this different from power functions?
A: In exponential functions, the variable is in the exponent (b^x), while in power functions it's in the base (xⁿ).

Q5: What's special about e as the base?
A: The natural base e (≈2.718) has unique mathematical properties, especially in calculus and continuous growth models.