Graphing Exponential Functions

Transcription

1 Graphing Eponential Functions Another coonly used type of non-linear function is the eponential function. We focus in this section on eponential graphs and solving equations. Another section is loaded with practical applications involving eponential functions. What distinguishes an eponential function fro a polynoial function is the placeent of the variable. With eponential functions, the variable is in the eponent. The general for for an eponential function is y = b, where b is a real nuber greater than 0 and not equal to. The not equal to constraint is due to the fact that y = is equivalent to the horizontal line given by the equation y = (a linear or constant function). The greater than 0 constraint is due to the ipossibility of finding a real square root (or any even root) of a negative nuber. For eaple, if we let b = 5 and try an value like, we have y = 5 = 5, which is undefined in the real nuber syste. Here are a few power rules, along with an eaple for each rule, as reinders. [Note: We assue that the values for a and b are real nubers, and the values for and n are integers.] Rule Eaple Rule Eaple a a n = a + n 5 5 = 5 6 n a = an b b n = = 6 8 a a = n a n = a n = 8 a 0 = 0 0 n = a = a n = a n = 8 a b n = a n b n = 9 b a n 5 = 5 = 5 b a n a n = an = = = = 8 6 We ll consider y = first with a table of values, along with a few probles showing the process y / 6 or / 8 or 0.5 / or 0.5 / or = = 6 = = = = = = = 8 6 = = 6

2 We can also raise the base to fractional powers and decial powers; for eaple, This work gives us ore points (0.5,.),.6,.78, and ( 5., 0.0) for the graph. Plotting these points and a continuous curve through the yields the following curve. There are ain features to notice about the graph of y = : This is an increasing function over its entire doain (of all real nubers). Notice that b is. Eponential functions increase whenever b >. The graph is a sooth and continuous curve, increasing draatically over its doain. For fun, raise to various powers and see how large the results are (then see what will overflow the calculator). The curve has y-intercept (0, ), since 0 =. In general, any nuber raised to the 0 power is. The graph approaches the -ais (the line y = 0) as gets saller and saller. The line given by y = 0 is the horizontal asyptote for this eponential curve. Continuing the thee of graphing eponential functions of the for y = b, with b > 0, consider the graphs of y = 0 and y =.5 below. y = 0

3 In general, with eponential functions of the for y = b, when b >, the function is an increasing function over its doain of all real nubers; the curve has y-intercept (0, ); and the line y = 0 is its horizontal asyptote. Now we ll consider the general case when 0 < b <. More specifically, we ll first let b =. The eponential function y = eaples, as well. This equation is equivalent to y =. contains the following points, and we show the process for a few / or 0.5 / or 0.5 / 8 or 0.5 / 6 or y = = = 0.5 = = = 6 = Plotting the points in our table and a continuous curve through these points gives a graph like the one shown to the right. Also copare both the tables and the graphs for y = and y =. These functions are irror reflections of one another over the y-ais. y = Two of the ain features of this curve are the sae as the ones we discussed for y =. This is a decreasing function over its entire doain (of all real nubers). [This is true whenever 0 < b <.] The curve has y-intercept (0, ), since raised to the 0 power is. The graph approaches the -ais (the line y = 0) as increases. The line given by y = 0 is a horizontal asyptote for the curve.

4 Just as y = and y = true for y = 0 and y = are irror reflections of one another over the y-ais, the sae is 0, as we see in the graph below. y = 0 y = 0 y = 0 In general, with eponential functions of the for y = b, when 0 < b <, the function is a decreasing function over its doain of all real nubers; the curve has y-intercept (0, ); and the line y = 0 is its horizontal asyptote. The nuber e: One of the ost interesting and useful nubers in any branches of science and atheatics is the nuber e. It pops up in several natural settings and several business applications. We ll see it in population growth and copound interest applications in the net section. The nuber e is an irrational nuber, very uch like the nuber π (which is approiately.59). They are fairly close to one another on the real nuber line. e The nuber e is defined as the nuber that the epression + approaches as gets larger and larger, approaching infinity. Using calculus notation, as, e. Consider the charts below. The first foreshadows copound interest applications

5 The second chart has continue to approach infinity. + +,000,000,000,000,000, ,000,000,000,000,000,000 Notice the trend in these answers; as increases, the value of + approaches a certain nuber. This nuber, naed e, is approiately It is an irrational nuber, neither repeating nor terinating as a decial. The graph of y = e would then be sandwiched between the graphs of y = and y =. Since the base is greater than, this is an increasing function over its doain of all real nubers. The curve has y-intercept (0, ), because any nuber (including the nuber e) raised to the 0 power is. Like the rest, it has the -ais (y = 0) as its horizontal asyptote. y = e Siilarly, the graph of y = e = e would be sandwiched between the graphs of y = and y =. Here, since the base is less than, this is a decreasing function over its doain of all real nubers. The curve also has y-intercept (0, ). Like the rest, it has the -ais (y = 0) as its horizontal asyptote. y = e = e Let s consider the eponential function given by y =. The for y = a b is considered a ore general for for an eponential function. Here, a is, and b is. Points on the graph include (, 0.75), (,.5), (0, ), (, 6), (, ), and so on: 0 5 y

6 You ay recognize a pattern in the y-coordinates. Notice fro both the table and the graph that the curve has y-intercept (0, ); otherwise the curve is siilar to y = in ters of rate of increase and in ters of its horizontal asyptote (y = 0). y = 0 The for y = a b is considered a ore general for for an eponential function. Until the function y =, the value of a has been on every eaple. In general, (0, a) represents the y-intercept of the graph. The value of a affects a vertical stretch of the curve. The value of b affects whether the graph increases or decreases and general steepness of the graph. All the transforations also apply to these general building block eponential functions. So, y = would be the graph of y = shifted left 5 units, reflected over the -ais, and then shifted down 6 units. For ore on these and other transforations, revisit Section.. One final technology note: When using the graphing calculator for eponential regression ( EpReg ), the calculator coputes the constants a and b for the function y = a b. The eponential regression for used by Microsoft Ecel is y = ae k, and the coputer software calculates the constants a and k. Coparing calculator and coputer regression equations, the values for a should be identical, and the values for b and k are very clearly related. The constant k is known as the growth rate (if k > 0) or the rate of decay (if k < 0). Eercises: For # 6, create a table of values and graph the given eponential functions. Be sure to show any intercepts and asyptotes in your graph.. y =. y =. y = 7 5. y =. y = y = 5 6

7 Describe in a sentence the relationship between the graph of y = and the graph of each of the following eponential functions. 7. y = + 8. y = + 9. y = 0. Find any - and y-intercepts of the graph of y = +.. Find any - and y-intercepts of the graph of y = +.. Using transforations, sketch the graph of y =.. Using transforations, sketch the graph of y = If each of the tables below represent points on the graphs of eponential functions, find the issing table values and the equation for each eponential function. (a) *(b) *(c) y y y List features that graphs of the following eponential functions have in coon. y = 5, y = 00, y = 0.00, and y = 5 6 *6. Find the reainder when 65 is divided by. [Hint: Use saller powers of and look for a pattern in the reainders.] *7. Find the reainder when 7 is divided by. [Hint: Use saller powers of and look for a pattern in the reainders.] *8. Siplify the following epression: and denoinator.] [Hint: Factor out the GCD for the nuerator