Exponential Graphs: Foundations

[Video: Introduction to Exponential Graphs]

Learn how to interpret and analyze exponential graphs, including their key features, growth vs. decay, and transformations.

Understanding Exponential Functions

Exponential functions have the form f(x) = a·b^x + c, where:

a is the initial value (y-intercept when c=0)
b is the growth/decay factor (b > 0, b ≠ 1)
c is the vertical shift

[Graph showing exponential growth and decay curves]

Figure 1: Exponential growth (b > 1) vs. decay (0 < b < 1)

Key Features of Exponential Graphs

Characteristics

Domain: All real numbers
Range: y > c (if a > 0) or y < c (if a < 0)
Asymptote: y = c (horizontal line)
y-intercept: (0, a + c)
Growth rate: Determined by base b

Growth vs. Decay

Exponential Growth

Occurs when b > 1

Graph rises from left to right
Common applications: population growth, compound interest

Example: f(x) = 2^x

Doubles with each unit increase in x

Exponential Decay

Occurs when 0 < b < 1

Graph falls from left to right
Common applications: radioactive decay, depreciation

Example: f(x) = (½)^x

Halves with each unit increase in x

Transformations of Exponential Graphs

General Form: f(x) = a·b^x-h + k

a: Vertical stretch/compression and reflection
h: Horizontal shift
k: Vertical shift (asymptote becomes y = k)

Example: f(x) = 3·2^x-1 - 4

• Initial value (when x=1): 3·2⁰ - 4 = -1

• Asymptote: y = -4

• Growth factor: 2 (doubles with each unit increase in x)

Key Takeaways

Identify growth vs. decay by the base value (b)
Always locate the horizontal asymptote first
Transformations work similarly to other functions
Real-world applications often involve exponential models
On SAT, focus on interpreting graphs in context

Practice Question

The graph of an exponential function f passes through the points (0, 5) and (1, 10). Which of the following could be the equation of f?

A) f(x) = 5·(½)^x

B) f(x) = 10·(½)^x

C) f(x) = 5·2^x

D) f(x) = 10·2^x

Follow these steps:

Use the point (0,5) to determine the initial value (a)
Use the point (1,10) to determine the growth factor (b)
Check which option matches both conditions

Step-by-Step Solution:

1. General form: f(x) = a·b^x

2. Using (0,5): 5 = a·b⁰ → a = 5

3. Using (1,10): 10 = 5·b¹ → b = 2

4. Therefore, f(x) = 5·2^x

The correct answer is C) f(x) = 5·2^x.