Vocabulary
Exponential function- A function of the form y=ab^x, where a≠0, b>0, and b≠1.
Exponential growth function- If a>0 and b>1, then the function y=ab^x is an exponential growth function with growth factor b.
Growth factor- The quantity b in the exponential growth function y=ab^x with a>0 and b>1.
Asymptote- A line that a graph approaches more and more closely.
Exponential growth function- If a>0 and b>1, then the function y=ab^x is an exponential growth function with growth factor b.
Growth factor- The quantity b in the exponential growth function y=ab^x with a>0 and b>1.
Asymptote- A line that a graph approaches more and more closely.
Essential Information
An exponential function has the form y=ab^x where a≠0 and the base b is a positive number other than 1. If a>0 and b>1, then the function y=ab^x is an exponential growth function, and b is called the growth factor. The simplest type of exponential growth function has the form y=b^x.
Parent Function for exponential Growth Functions
The function f(x)=b^x, where b>1, is the parent function for the family of exponential growth functions with base b.
Parent Function for exponential Growth Functions
The function f(x)=b^x, where b>1, is the parent function for the family of exponential growth functions with base b.
The domain of f(x)=b^x is all real numbers. The range is y>0.
The graph of a function y=ab^x is a vertical stretch or shrink of the graph of y=b^x. The y-intercept of the graph of y=ab^x occurs at (0,a) rather than (0,1)
The graph of a function y=ab^x is a vertical stretch or shrink of the graph of y=b^x. The y-intercept of the graph of y=ab^x occurs at (0,a) rather than (0,1)
Translations
A function in the form y=ab^(x-h)+k can be graphed by sketching the graph of y=ab^x. Then translate the graph horizontally by h units and vertically k units.
A function in the form y=ab^(x-h)+k can be graphed by sketching the graph of y=ab^x. Then translate the graph horizontally by h units and vertically k units.
Exponential Growth Models
When a real- life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:
y=a(1+r)^t
Where a is the initial amount and r is the percent increase expressed as a decimal. 1+r is the growth factor.
Compound Interest
Exponential growth functions are used in real-life situations involving compound interest. Compound interest is interest paid on the initial investment, called the principal, and on previously earned interest. Interest paid only on the principal is called simple interest.
P: initial principal P deposited in an account
r: annual rate (expressed as a decimal)
n: the number of times compounded each year
t: years
A: the amount in the account after t years
A=P(1+ (r/n))^nt
When a real- life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:
y=a(1+r)^t
Where a is the initial amount and r is the percent increase expressed as a decimal. 1+r is the growth factor.
Compound Interest
Exponential growth functions are used in real-life situations involving compound interest. Compound interest is interest paid on the initial investment, called the principal, and on previously earned interest. Interest paid only on the principal is called simple interest.
P: initial principal P deposited in an account
r: annual rate (expressed as a decimal)
n: the number of times compounded each year
t: years
A: the amount in the account after t years
A=P(1+ (r/n))^nt