3.1: Exponential Functions and Their Graphs

Exponential Functions:

f(x) = a^x

where a is greater than 0, not equal to one, and x is any real number





Exponential Graphs:

The graphs of all exponetial functions have similar characteristics:

• Domain: (-∞, ∞)
• Range: (0, ∞)
• y-intercept: (0, 1)
• No x-intercept
• Horizontal Asymptote of y=0
• Continuous

Note:

• A positive x indicates an increasing function
• A negative x indicates a decreasing function

Transformations of Exponential Graphs:
y = ab^(x-c) + d

• a -> causes the graph to vertically stretch or shrink; affects the y-intercept; if negative, causes a reflection in the x-axis
• c -> causes the graph to move left or right (left if +, right if -)
• d -> causes the graph to move up and down
• x -> a negative x causes the graph to reflect in the y-axis

The Natural Base e:

• e = 2.71828...
• e is called the natural base and its function is called the natural exponential function

f(x) = e^x

• Domain: (-∞, ∞)
• Range: (0, ∞)
• y-intercept: (0,1)

Rules of Exponents:

• a^x * a^y = a^(x+y)
• a^x / a^y = a^(x-y)
• a^-x = 1 / a^x

Compound Interest:

• Continuously Compounded Interest: A = Pe^rt
• Discretely Compounded Interest: A = P(1 + r/n)^nt
• A = present value
• P = principle value (how much money is invested)
• e = natural base
• r = annual rate of return *must convert into decimal form
• t = time in years
• n = number of compounding periods per year (ex. quarterly = 4, monthly = 12)