EXPONENTIAL FUNCTION

Exponential Function

Definition Of Exponential Function

An Exponential Function is a function of the form y = ab^x, where both a and b are greater than 0 and b is not equal to 1.

More About Exponential Function

Exponential Decay
Exponential decay occurs when a quantity decreases by the same proportion r in each time period t. If A₀ is the initial amount, then the amount at time t is given by A = A₀(1 - r)^t, where r is called the decay rate, 0 < r="">< 1,and="" (1-r)="" is="" called="" the="" decay="">
Exponential Growth
Exponential growth occurs when a quantity increases by the same proportion r in each time period t. If A₀ is the initial amount then the amount at time t is given by A = A₀(1 - r)^t, where r is called the growth rate, 0 < r="">< 1,="" and="" (1="" +="" r)="" is="" called="" the="" growth="" factor.="">

Video Examples: Introduction To Exponential Functions

Example of Exponential Function

y = 4.3(1.23)^X is an exponential function.

Solved Example on Exponential Function

Ques: Evaluate the exponential function y = 5(4)^x when x = 2.5. Round off the answer to the nearest hundredth.

Choices:

A. 160
B. 625
C. 140
D. 80.58
Correct Answer: A

Solution:

Step 1: y = 5(4)^x [Original exponential function.]
Step 2: y = 5(4)^2.5[Replace x with 2.5.]
Step 3: 5(32) = 160 [Use Calculator.]

Real World Connections for Exponential Function

Exponential functions are used in banking and finance to calculate compound interest.
Radioactive decay, population growth - these can be modeled using exponential functions.

Quick Summary

Exponential functions model growth or decay.
The base 'b' determines if the function represents growth (b > 1) or decay (0 < b < 1).
The initial value is represented by 'a'.

\[ y = ab^x \]

⚠️ Common Mistakes

Confusing exponential functions with polynomial functions.
Incorrectly applying the exponent to the coefficient 'a'.
Misunderstanding the impact of the base 'b' on growth or decay.

🌍 Real-World Uses

Calculating compound interest on a bank account, where the balance grows exponentially over time.
Modeling the population growth of bacteria in a petri dish, assuming unlimited resources.
Determining the radioactive decay of a substance like Carbon-14, used in carbon dating.

📋 Standards Alignment

CCSS.MATH.HSF.LE.A.1
CCSS.MATH.HSF.LE.A.2

🍎 Teacher Insights

Use real-world examples like compound interest and population growth to illustrate exponential functions. Emphasize the difference between linear and exponential growth.

🎓 Prerequisites

Basic Algebra
Understanding of variables and exponents

Check Your Knowledge

Q1: Which of the following is an exponential function?

A. y = 3x + 2 B. y = x^2 C. y = 2^x D. y = x^3 + 1

Q2: What does 'a' represent in the exponential function y = ab^x?

A. The growth rate B. The decay rate C. The initial value D. The exponent

Frequently Asked Questions

Q: What is the difference between exponential growth and exponential decay?
A: Exponential growth occurs when the quantity increases by a constant factor in each time period (b > 1). Exponential decay occurs when the quantity decreases by a constant factor in each time period (0 < b < 1).

Q: How can I identify an exponential function from a table of values?
A: Look for a constant ratio between consecutive y-values for equally spaced x-values.