Exponential Functions

📌 What is an Exponential Function?

An exponential function is a function where the variable appears in the exponent:

$ f(x) = ab^x $

Where:

$ a $ = initial value (y-intercept when $ x = 0 $)
$ b $ = base (growth/decay factor), $ b > 0, b \neq 1 $
$ x $ = exponent (independent variable)

Domain and Range of Exponential Functions

General Form: $ f(x) = ab^x + k $

Domain:

All real numbers: $ (-\infty, \infty) $

Exponential functions are defined for all values of $ x $

Range:

If $ a > 0 $: $ y > k $ or $ (k, \infty) $
If $ a < 0 $: $ y < k $ or $ (-\infty, k) $

The range depends on the horizontal asymptote $ y = k $

📝 Examples - Domain and Range:

Example 1: $ f(x) = 2^x $

Domain: $ (-\infty, \infty) $
Range: $ (0, \infty) $ or $ y > 0 $
Asymptote: $ y = 0 $

Example 2: $ f(x) = 3 \cdot 2^x - 5 $

Domain: $ (-\infty, \infty) $
Range: $ (-5, \infty) $ or $ y > -5 $
Asymptote: $ y = -5 $

Evaluating Exponential Functions

How to Evaluate:

Substitute the given value for $ x $
Calculate the power (base raised to the exponent)
Multiply by the coefficient if present
Add/subtract any constant term

📝 Examples - Evaluating:

Example 1: Evaluate $ f(x) = 2^x $ when $ x = 3 $

$ f(3) = 2^3 = 8 $

Example 2: Evaluate $ f(x) = 5 \cdot 3^x $ when $ x = 2 $

$ f(2) = 5 \cdot 3^2 = 5 \cdot 9 = 45 $

Example 3: Evaluate $ f(x) = 100 \cdot (0.5)^x $ when $ x = 4 $

$ f(4) = 100 \cdot (0.5)^4 = 100 \cdot 0.0625 = 6.25 $

Example 4: Evaluate $ f(x) = 2 \cdot 3^x + 1 $ when $ x = 0 $

$ f(0) = 2 \cdot 3^0 + 1 = 2 \cdot 1 + 1 = 3 $

Exponential Growth and Decay

Exponential Growth: $ b > 1 $

$ f(x) = a(1 + r)^x $ or $ f(x) = ab^x $ where $ b > 1 $

$ a $ = initial amount
$ r $ = growth rate (as a decimal)
$ x $ or $ t $ = time
Growth factor: $ b = 1 + r $
Function increases as $ x $ increases

Exponential Decay: $ 0 < b < 1 $

$ f(x) = a(1 - r)^x $ or $ f(x) = ab^x $ where $ 0 < b < 1 $

$ a $ = initial amount
$ r $ = decay rate (as a decimal)
$ x $ or $ t $ = time
Decay factor: $ b = 1 - r $
Function decreases as $ x $ increases

📝 Example - Growth:

A population of 500 bacteria doubles every hour. Write the exponential function and find the population after 3 hours.

Step 1: Identify values

Initial amount: $ a = 500 $
Doubles means: $ b = 2 $ (growth factor)

Step 2: Write function

$ P(t) = 500 \cdot 2^t $

Step 3: Find population after 3 hours

$ P(3) = 500 \cdot 2^3 = 500 \cdot 8 = 4000 $ bacteria

📝 Example - Decay:

A car worth $30,000 depreciates at 15% per year. Write the function and find its value after 5 years.

Step 1: Identify values

Initial value: $ a = 30000 $
Decay rate: $ r = 0.15 $
Decay factor: $ b = 1 - 0.15 = 0.85 $

Step 2: Write function

$ V(t) = 30000(0.85)^t $

Step 3: Find value after 5 years

$ V(5) = 30000(0.85)^5 \approx 30000(0.4437) \approx $13,311 $

Graphing Exponential Functions

Key Features of Exponential Graphs:

For $ f(x) = ab^x + k $:

Y-intercept: $ (0, a + k) $ — substitute $ x = 0 $
Horizontal asymptote: $ y = k $
Domain: All real numbers
If $ b > 1 $: Growth (increases left to right)
If $ 0 < b < 1 $: Decay (decreases left to right)
Graph never touches the asymptote
Always positive if $ a > 0 $ and above asymptote

Steps to Graph:

Identify the horizontal asymptote $ y = k $
Find the y-intercept by substituting $ x = 0 $
Calculate 2-3 more points by choosing convenient $ x $ values
Determine if it's growth ($ b > 1 $) or decay ($ 0 < b < 1 $)
Draw the asymptote, plot points, and sketch the curve

📝 Example - Graphing:

Graph $ f(x) = 2 \cdot 3^x - 1 $

Step 1: Horizontal asymptote

$ y = -1 $

Step 2: Y-intercept

$ f(0) = 2 \cdot 3^0 - 1 = 2 \cdot 1 - 1 = 1 $ → Point: $ (0, 1) $

Step 3: Additional points

$ f(1) = 2 \cdot 3^1 - 1 = 6 - 1 = 5 $ → $ (1, 5) $
$ f(-1) = 2 \cdot 3^{-1} - 1 = \frac{2}{3} - 1 = -\frac{1}{3} $ → $ (-1, -\frac{1}{3}) $

Step 4: Type

Since $ b = 3 > 1 $, this is exponential growth

Step 5: Draw asymptote at $ y = -1 $, plot points, and sketch increasing curve

Transformations of Exponential Functions

Standard Form:

$ f(x) = a \cdot b^{x-h} + k $

$ a $: Vertical stretch/compression and reflection
$ h $: Horizontal shift (right if $ h > 0 $, left if $ h < 0 $)
$ k $: Vertical shift (up if $ k > 0 $, down if $ k < 0 $)
Horizontal asymptote moves to $ y = k $

Identifying Linear vs. Exponential Functions

From a Table of Values:

Linear Function:

Constant difference in $ y $-values for equal $ x $ intervals
Form: $ y = mx + b $

Exponential Function:

Constant ratio in $ y $-values for equal $ x $ intervals
Form: $ y = ab^x $

📝 Example - Identify from Table:

$ x $	0	1	2	3
$ y $	3	6	12	24

Check differences: $ 6-3=3, 12-6=6, 24-12=12 $ → Not constant (not linear)
Check ratios: $ \frac{6}{3}=2, \frac{12}{6}=2, \frac{24}{12}=2 $ → Constant!
Answer: Exponential with $ a = 3, b = 2 $ → $ y = 3 \cdot 2^x $

Exponential Functions Over Unit Intervals

Key Concept:

For exponential functions, consecutive integer values of $ x $ (unit intervals) produce $ y $-values with a constant ratio.

$ \frac{f(x+1)}{f(x)} = b $ (constant for all $ x $)

This ratio equals the base $ b $ of the exponential function.

Word Problems with Exponential Functions

Common Formulas:

1. Compound Interest:

$ A = P(1 + r)^t $

$ P $ = principal, $ r $ = rate, $ t $ = time, $ A $ = amount

2. Population Growth:

$ P(t) = P_0(1 + r)^t $

3. Radioactive Decay:

$ N(t) = N_0 \left(\frac{1}{2}\right)^{t/h} $

$ h $ = half-life

4. General Growth/Decay:

$ y = a(1 \pm r)^t $

Use + for growth, − for decay

📝 Example - Compound Interest:

You invest $5,000 at 6% annual interest compounded yearly. How much will you have after 10 years?

Given:

$ P = 5000 $
$ r = 0.06 $
$ t = 10 $

Formula:

$ A = 5000(1.06)^{10} $

Calculate:

$ A = 5000(1.790847) \approx $8,954.24 $

📝 Example - Population Growth:

A town's population is 20,000 and grows at 3% per year. Write the function and find the population after 15 years.

Write function:

$ P(t) = 20000(1.03)^t $

After 15 years:

$ P(15) = 20000(1.03)^{15} \approx 20000(1.558) \approx 31,160 $ people

⚡ Quick Summary

Property	Exponential Function
General Form	$ f(x) = ab^x + k $
Domain	All real numbers
Range	$ y > k $ (if $ a > 0 $)
Asymptote	$ y = k $
Growth	$ b > 1 $
Decay	$ 0 < b < 1 $

Exponential functions have variable in the exponent
Growth: $ y = a(1+r)^t $, Decay: $ y = a(1-r)^t $
Linear has constant differences; exponential has constant ratios
Y-intercept at $ (0, a+k) $; horizontal asymptote at $ y = k $
Graph increases (growth) or decreases (decay) rapidly