Weve looked at linear and quadratic functions, polynomial functions and rational functions. We are now going to study a new function called exponential.
Presentation on theme: "Weve looked at linear and quadratic functions, polynomial functions and rational functions. We are now going to study a new function called exponential."— Presentation transcript:
Weve looked at linear and quadratic functions, polynomial functions and rational functions. We are now going to study a new function called exponential functions. They are different than any of the other types of functions weve studied because the independent variable is in the exponent. Lets look at the graph of this function by plotting some points. x 2 x /2 -2 1/4 -3 1/ Recall what a negative exponent means: BASE
The asymptote The asymptote is a line that the graph is heading towards but will never meet. At Studies these are usually horizontal, or vertical lines. The asymptote
Compare the graphs 2 x, 3 x, and 4 x Characteristics about the Graph of an Exponential Function where a > 1 What is the domain of an exponential function? 1. Domain is all real numbers What is the range of an exponential function? 2. Range is positive real numbers What is the x intercept of these exponential functions? 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 What is the y intercept of these exponential functions? 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing Are these exponential functions increasing or decreasing? 6. The x-axis (where y = 0) is a horizontal asymptote for x - Can you see the horizontal asymptote for these functions?
All of the transformations that you learned apply to all functions, so what would the graph of look like? up 3 up 1 Reflected over x axis down 1right 2
Reflected over y-axisThis 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 (well see some of these later this chapter). To model these we need to learn about a special base.
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. If a u = a v, then u = v 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 2s, We just used the property and equated the exponents. You could solve this for x now.
Lets try one more: The left hand side is 4 to the something but the right hand side cant 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:
Solving equations with exponentials G-SOLV ROOT G-SOLV ISCT