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Choices… • A wealthy man nearing the end of his life called his two children to his bedside. He wanted to leave them with the opportunity to experience the richness of life that he had enjoyed. • He offered them the choice of $1,000,000.00 cash or • $.01 cash (yes, ONE Penny) that would double everyday for one month (30 days). He then sent them home to consider the offer. • Which choice is better? Why? The Answer… Remember, you still have to add up everything for each day… 9.5 Notes I. Exploring 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 x x 2 x some points. 8 3 2 1 0 -1 -2 -3 8 4 2 1 1/2 1/4 1/8 f x 2 BASE Recall what a negative exponent means: 1 1 f 1 2 2 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 The box summarizes the general shapes of exponential function graphs. Graphs of Exponential Functions Compare the graphs 2x, 3x , and 4x Characteristics about the Graph of an Exponential x Function f x a 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 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 - f x 4 x f x 3x f x 2 x Can What What you is isthe the seerange x What Are these is the ythe of intercept horizontal domain an exponential of of these intercept exponential ofan these function? exponential asymptote exponential for exponential functions functions? these function? functions? functions? increasing or decreasing? II. Graphing Exponential Functions with a Table Ex 1: Graph y = 0.5(2)x. Choose several values of x and generate ordered pairs. x y = 0.5(2)x –1 0.25 0 0.5 1 1 2 2 Graph the ordered pairs and connect with a smooth curve. Ex 2: Graph y = 2x. Choose several values of x and generate ordered pairs. x –1 0 1 2 y = 2x 0.5 1 2 4 Graph the ordered pairs and connect with a smooth curve. • • • • Ex 3: Graph y = 0.2(5)x. Choose several values of x and generate ordered pairs. x –1 0 1 2 Graph the ordered pairs and connect with a smooth curve. • 0.2(5)x y= 0.04 0.2 1 5 • • • Ex 4: Choose several values of x and generate ordered pairs. 1 x x y =– (2) 4 –1 0 1 2 –0.125 –0.25 –0.5 –1 Graph the ordered pairs and connect with a smooth curve. • • • • Ex 5: Graph y = –6x. Choose several values of x and generate ordered pairs. x –1 0 1 2 y = –6x –0.167 –1 –6 –36 Graph the ordered pairs and connect with a smooth curve. • • • Ex 6: Graph y = –3(3)x. Choose several values of x and generate ordered pairs. x –1 0 1 2 y = –3(3)x –1 –3 –9 –27 Graph the ordered pairs and connect with a smooth curve. • • • Ex 7: Graph each exponential function. Choose several values of x and generate ordered pairs. x –1 0 1 2 Graph the ordered pairs and connect with a smooth curve. 1 y = –1( )x 4 –4 –1 –0.25 – 0.0625 • • • • Ex 8: Graph each exponential function. y = 4(0.6)x Choose several values of x and generate ordered pairs. x –1 0 1 2 y = 4(0.6)x 6.67 4 2.4 1.44 Graph the ordered pairs and connect with a smooth curve. • • • • Ex 9: Graph each exponential function. Choose several values of x and generate ordered pairs. x –1 0 1 2 Graph the ordered pairs and connect with a smooth curve. x1 y = 4( ) 4 16 4 1 0.25 • • • • Ex 10: Graph each exponential function. y = –2(0.1)x Choose several values of x and generate ordered pairs. x –1 0 1 2 y = –2(0.1)x –20 –2 –0.2 –0.02 Graph the ordered pairs and connect with a smooth curve. • • • • II. 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. Example You should get 2.718281828 for TI-83 f x e x f x 3x f x 2 x III. Identifying Exponential Functions From a Table Exponential functions have constant ratios. As the x-values increase by a constant amount, the y-values are multiplied by a constant amount. This amount is the constant ratio and is the value of b in f(x) = abx. y Ex 1: Tell whether each set of ordered pairs satisfies an exponential function. Explain your answer. {(0, 4), (1, 12), (2, 36), (3, 108)} This is an exponential function. As the x-values increase by a constant amount, the y-values are multiplied by a +1 constant amount. +1 +1 x 0 1 2 3 y 4 12 36 108 3 3 3 Ex 2: Tell whether each set of ordered pairs satisfies an exponential function. Explain your answer. {(–1, –64), (0, 0), (1, 64), (2, 128)} This is not an exponential function. As the x-values increase by a constant amount, the y-values are not multiplied by a constant amount. +1 +1 +1 x y –1 –64 0 0 1 64 2 128 + 64 + 64 + 64 Ex 3: Tell whether each set of ordered pairs satisfies an exponential function. Explain your answer. {(–1, 1), (0, 0), (1, 1), (2, 4)} This is not an exponential function. As the x-values increase by a constant amount, the y-values are not multiplied by a constant amount. +1 +1 +1 x –1 0 1 2 y 1 0 1 4 –1 +1 +3 Ex 4: Tell whether each set of ordered pairs satisfies an exponential function. Explain your answer. {(–2, 4), (–1 , 2), (0, 1), (1, 0.5)} This is an exponential function. As the x-values increase by a constant amount, the y-values are multiplied by a constant amount. +1 +1 +1 x y –2 4 –1 2 0 1 1 0.5 × 0.5 × 0.5 × 0.5 IV. Applications Ex 1: In 2000, each person in India consumed an average of 13 kg of sugar. Sugar consumption in India is projected to increase by 3.6% per year. At this growth rate the function f(x) = 13(1.036)x gives the average yearly amount of sugar, in kilograms, consumed per person x years after 2000. Using this model, in about what year will sugar consumption average about 18 kg per person? Enter the function into the Y = editor of a graphing calculator. Press . Use the arrow keys to find a y-value as close to 18 as possible. The corresponding x-value is 9. The average consumption will reach 18 kg in 2009. Ex 2: An accountant uses f(x) = 12,330(0.869)x, where x is the time in years since the purchase, to model the value of a car. When will the car be worth $2000? Enter the function into the Y = editor of a graphing calculator. Check It Out! Example 6 Continued An accountant uses f(x) = 12,330(0.869)x, is the time in years since the purchase, to model the value of a car. When will the car be worth $2000? Press . Use the arrow keys to find a y-value as close to 2000 as possible. The corresponding x-value is 13. The value of the car will reach $2000 after about year 13. Lesson Quiz: Part I Tell whether each set of ordered pairs satisfies an exponential function. Explain your answer. 1. {(0, 0), (1, –2), (2, –16), (3, –54)} No; for a constant change in x, y is not multiplied by the same value. 2. {(0,–5), (1, –2.5), (2, –1.25), (3, –0.625)} Yes; for a constant change in x, y is multiplied by the same value. Lesson Quiz: Part II 3. Graph y = –0.5(3)x. Lesson Quiz: Part III 4. The function y = 11.6(1.009)x models residential energy consumption in quadrillion Btu where x is the number of years after 2003. What will residential energy consumption be in 2013? 12.7 quadrillion Btu 5. In 2000, the population of Texas was about 21 million, and it was growing by about 2% per year. At this growth rate, the function f(x) = 21(1.02)x gives the population, in millions, x years after 2000. Using this model, in about what year will the population reach 30 million? 2018