Exponential functions 3.1

Exponential functions – functions in which the (independent) variable is in the exponent. Let’s create a graph of this function by plotting some points. BASE x 2x 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 3 8 2 4 Recall : 1 2 0 1 -1 1/2 -2 1/4 -3 1/8

Formal definition of the Exponential Function The exponential function f with the base b is defined by or , where b is a positive constant other than 1 ( ) and x is any real number. Why is the base value restricted?

Compare the graphs y=2x, y=3x, and y=4x Characteristics of the Graph of an Exponential Function (parent) Case 1: , 1. Domain is all real numbers: (- , + ) 2. Range is positive real numbers: (0, + ) There are no x intercepts because there is no x value that you can put in the function to make it = 0 Can you see the horizontal asymptote for these functions? What is the x intercept of these exponential functions? What is the range of an exponential function? What is the domain of an exponential function? What is the y intercept of these exponential functions? Are these exponential functions increasing or decreasing? The y intercept is always (0,1) because b 0 = 1 5. The graph is always increasing The x-axis (where y = 0) is a horizontal asymptote for x  - 

All of the transformations that you learned apply to all functions, so what would the graph of look like? up 3 right 2 down 1 Reflected over the x-axis up 1

Reflected about y-axis This equation could be rewritten in a different form: This is Case 2: , and So if the base of our exponential function is between 0 and 1, the function is decreasing. It will have the same domain, range, intercepts, and asymptote. There are many occurrences in nature that can be modeled with an exponential function. To model these we need to use the natural base.

The Base “e” (also called the natural base) Ask your calculator to find e1. You do this by using the ex button (generally you’ll need to hit the 2nd or blue button first to get it depending on the calculator). After hitting the ex, you then enter the exponent you want (in this case 1) and push = or enter. If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the ex. You should get 2.718281828 How is number e defined? Example for TI-84 Plus

Solving simple exponential equations using property: If bm = bn, then m = n That is: if we have exponential functions on both sides of an equation, and if we can write both sides of this equation using the same base, then the exponents are equal. Re-write the right side with the base 2 Use the property above. The bases are both 2 so the exponents must be equal. We did not cancel the 2’s, We just used the property and equated the exponents. Solve the equation.

Example: Re-write both the left and right hand sides with the same base 2 (as 2 to the some power). Use properties of exponents Now that both sides are written with the same base, set exponents equal to each other. Check:

Some applications of Exponential Functions Compound Interest ...