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Objectives: 1. To graph exponential functions 2. To solve simple exponential equations 3. To find an exponential growth or decay model Assignment: • P. 226: 11-16 (Some) – By hand • • • • • • P. 226: 17-22 (Some) P. 227: 39-44 (Some) P. 227: 45-58 P. 227: 68 (a,b) P. 228: 71, 73, 75, 80 Homework Supplement: 1-3 The fictional Gordon Moore Intelligence Reward Organization (GMIRO) has decided to reward one clever and assiduous student by giving them a substantial amount of money. This student must make one of 2 choices: Gordon Moore: Richer than you. 1. GMIRO gives the student a penny on the first day. On the second day, GMIRO will double the penny. On the third day, GMIRO will double the money from the previous day. GIMRO will continue to do this until the end of the month when they will give you the final sum of money. Gordon Moore: Richer than you. 2. GMIRO gives the student one million dollars. As a potential candidate for GMIRO’s reward, which option would you choose? Gordon Moore: Richer than you. The number of transistors on an integrated circuit (IC) doubles every 2 years. • N2 = 2(y2 – y1)/2·N1 ©Pearson Prentice Hall, 2004 Consider the algebraic expression 3(2x + y). We usually think of this as 3 distributed through the parenthesis, but it also means 3 copies of 2x + y: 2x + y + 2x + y + 2x + y Multiplication is simply repeated addition. Now consider the algebraic expression (2x + y)3. What could be done to the illustration below to represent this new expression? (2x + y) x (2x + y) x (2x + y) An exponent is simply repeated multiplication. Simplify the following expressions: 5 m 3 m Here’s a few good properties: Multiplication Property of Exponents Power Properties of Exponents bm·bn = bm+n b = nm b m (ab)n = anbn b (bm)n bmn Division Property of Exponents n Simplify the following expressions: 32 74 4 74 3 And here’s a couple more: Negative Exponents b n 1 n b 1 n b n b Zero Exponents b0 = 1 (b ≠ 0) Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents. a a x 2 2 a Let a x a a So a1 2 a a2x a a 2 x a1 2x 1 1 x 2 Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents. c n c x n n c Let c x n c c So c1 n n c c nx c c nx c1 nx 1 1 x n Without a calculator, evaluate the following. You will be able to graph exponential functions An exponential function with base b is denoted as f (x) = bx, where b > 0 and b ≠ 1. If b = 1, then we’d have y = 1x. It’s constant. Which is just y = 1. And that’s not exponential. Without a calculator, graph each of the following: 1. f (x) = 2x 2. g (x) = 3x 3. h (x) = 4x Graph of y = bx, b > 1 Domain: (−∞, ∞) Range: (0, ∞) Horizontal asymptote: y = 0 y-intercept: 1 Increasing (growth) Continuous One-to-one Without a calculator, graph each of the following: 1. f (x) = 2−x 2. g (x) = 3−x 3. h (x) = 4−x Graph of y = b-x, b > 1 Domain: (−∞, ∞) Range: (0, ∞) Horizontal asymptote: y = 0 y-intercept: 1 Decreasing (decay) Continuous One-to-one How you learned it last year: y 2 x 1 y x 2 1 y 2 x Also notice that the graph of y = b−x is the same as y = bx, just reflected over the y-axis. Use the following GSP demo to discover the roll of a, h, and k in the exponential function: y a b xh k Identify the parent for each function, then state the SRT transformations to make each graph. Finally, state the domain and range of each. 1. y = 4x – 2 2. y = −(1/2)‧3x 3. y = 2−x + 3 You will be able to solve simple exponential equations Horizontal Line Test One-to-One Every input has 1 output Every output has 1 input Enables us to solve simple exponential equations when the bases are equal. If 2x = 23, then what is the value of x? Property of Equality of Exponential Equations If b is a positive real number not equal to 1, then bx = by iff x = y. – 2x = 23 iff x = 3 – This means that one way to solve exponential equations is to make the bases equal so that the exponents are equal. Solve: You will be able to find an exponential growth or decay model Often in a problem, you have to add a percentage of a price back to the price, like taxes and tips. To do this, add one to the percent, and then multiply. x 8%x x .08x x 1 .08 x 1.08 $5.50 8% $5.50 $5.50 .08 $5.50 $5.50 1 .08 $5.50 1.08 In many real-life situations, a quantity will grow at a constant rate. This is like the previous tip/tax problem, except it is done over and over again. $5.50 1.081.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 $5.50 1.08 t In a real-life situation, when a quantity a continues to increased by a fixed percent r each year (or some other time frame), the amount y of the quantity after time t can be modeled by: y a 1 r t Growth Factor According to some website, the population of Denton, TX was 119,454 in 2008. This was up from 3.01% from the previous year. If the population continues to grow at this rate, what do you expect the population of Denton to be in 2020? Often in a problem, you have to subtract a percentage of a price from the price, like discounts. To do this, subtract the percent from one, and then multiply. x 10%x x .1x x 1 .1 x .9 $5.50 10% $5.50 $5.50 .1 $5.50 $5.50 1 .1 $5.50 .9 In many real-life situations, a quantity will shrink at a constant rate. This is like the previous discount problem, except it is done over and over again. $5.50 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 . $5.50 .9 t In a real-life situation, when a quantity a continues to decrease by a fixed percent r each year (or some other time frame), the amount y of the quantity after time t can be modeled by: y a 1 r t Decay Factor A new car costs $25,000. The value of the car decreases by 15% each year. Write an exponential decay model giving the car’s value y (in dollars) after t years. Estimate the value after 2 years. Assume the car from the previous example continues to decrease in value by 15% every year. In how many years with the value of the car be ½ the original value? The MSRP on my 2001 Honda Civic LX was $14,810. Currently, the Kelley Blue Book Price for the same car averages $4,913. Write an exponential decay model for the value V of my car t years after 2001. Then find the year when my car will be worth a measly $1,000. Objectives: 1. To graph exponential functions 2. To solve simple exponential equations 3. To find an exponential growth or decay model Assignment: • P. 226: 11-16 (Some) – By hand • • • • • • P. 226: 17-22 (Some) P. 227: 39-44 (Some) P. 227: 45-58 P. 227: 68 (a,b) P. 228: 71, 73, 75, 80 Homework Supplement: 1-3