exponential functions

Let’s examine exponential functions Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent. Let’s look at the graph of this function by plotting some points. x 2x 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 3 8 2 4 BASE 1 2 0 1 Recall what a negative exponent means: -1 1/2 -2 1/4 -3 1/8

Definition of Exponential Functions The exponential function f with a base b is defined by f(x) = bx where b is a positive constant other than 1 (b > 0, and b ≠ 1) and x is any real number. So, f(x) = 2x, looks like:

Compare the graphs 2x, 3x , and 4x Characteristics about the Graph of an Exponential Function where a > 1 1. Domain is all real numbers 2. Range is positive real numbers 3. 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? 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing 6. 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 x axis up 1

Reflected about y-axis This equation could be rewritten in a different form: So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be 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 learn about a special base.

Graphing Exponential Functions Four exponential functions have been graphed. Compare the graphs of functions where b > 1 to those where b < 1

Graphing Exponential Functions So, when b > 1, f(x) has a graph that goes up to the right and is an increasing function. When 0 < b < 1, f(x) has a graph that goes down to the right and is a decreasing function.

Behaviors of Exponential Functions The domain of f(x) = bx consists of all real numbers (-, ). The range of f(x) = bx consists of all positive real numbers (0, ). The graphs of all exponential functions pass through the point (0,1). This is because f(o) = b0 = 1 (bo). The graph of f(x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. f(x) = bx is one-to-one and has an inverse that is a function.

Transformations Vertical translation f(x) = bx + c Shifts the graph up if c > 0 Shifts the graph down if c < 0

Transformations Horizontal translation: g(x)=bx+c Shifts the graph to the left if c > 0 Shifts the graph to the right if c < 0

Transformations Reflecting g(x) = -bx reflects the graph about the x-axis. g(x) = b-x reflects the graph about the y-axis.

Transformations Vertical stretching or shrinking, f(x)=cbx: Stretches the graph if c > 1 Shrinks the graph if 0 < c < 1

Transformations Horizontal stretching or shrinking, f(x)=bcx: Shinks the graph if c > 1 Stretches the graph if 0 < c < 1

The Base “e” (also called the natural base) To model things in nature, we’ll need a base that turns out to be between 2 and 3. Your calculator knows this base. Ask your calculator to find e1. You do this by using the ex button (generally you’ll need to hit the 2nd or yellow 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 Example for TI-83

If au = av, then u = v This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal. The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something? Now we 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. You could solve this for x now.

Let’s try one more: The left hand side is 4 to the something but the right hand side can’t be written as 4 to the something (using integer exponents) We could however re-write both the left and right hand sides as 2 to the something. So now that each side is written with the same base we know the exponents must be equal. Check: