Understanding Exponential Functions: Graphing and Real-life Applications
TLDR; Exponential functions are graphed and analyzed, with examples of growth, decay, symmetry, and real-life applications.
📈 Exponential Functions Definition
An exponential function is given by f(x) = b^x, where x is any real number and b is greater than 0 but not equal to 1.
The base 'b' is called the base of the exponential function and it is called exponential because the variable is in the position of the exponent.
When the base 'b' is greater than 1, the function f(x) is a positive increasing continuous function, known as an exponential growth function.
On the other hand, if 'b' is between 0 and 1, the function f(x) is a positive decreasing continuous function, known as an exponential decay function.
The function is always positive, meaning always above the x-axis, and always has a horizontal asymptote of y = 0 if there are no translations to the graph.
📊 Graphing Exponential Function
When graphing an exponential function like f(x) = 2^x, a T-table is used to find the corresponding y values.
For example, when x is 0, y is 1; when x is 1, y is 2, and so on.
The graph of an exponential growth function goes up from left to right and has a y-intercept of (0,1).
It also has a horizontal asymptote of y = 0, increases over its entire domain, and is concave up.
For an exponential decay function like f(x) = 1/2^x, the graph goes downhill from left to right, has a y-intercept of (0,1), a horizontal asymptote of y = 0, and is decreasing over its entire domain, while being concave up.
🔄 Y-axis Symmetry
The function f(x) = 2^-x is shown to be equivalent to f(x) = 1/2^x, demonstrating y-axis symmetry.
When comparing the graphs of f(x) = 2^-x and f(x) = 2^x, it is evident that these two functions have y-axis symmetry.
🌍 Real-life Applications
Exponential functions have real-life applications such as population growth, bacteria growth, internet users, medication dosage, and radioactive decay.
The speaker mentions a website that provides information on world population and its changes over time.
The website, intmath.com, displays the changing world population in real-time and provides data on the rate of increase, how many people are added to the planet each second, day, and year.
It also allows users to visualize population changes over different years.
👩🏫 Outro
The video provides a quick overview of graphing exponential functions and some of their basic properties.
The next topic to be covered is defining derivatives of exponential functions.