LINEAR AND EXPONENTIAL FUNCTION

Transcription

1 LINEAR AND EXPONENTIAL FUNCTION At the end of this workbook you should Be able to identify when data have linear or exponential behavior Be able to find the equations of linear functions and exponential functions from given information Interpret the slope of the line in solving problems Understand what growth (decay) factor is Distinguish between increasing/decreasing linearly (by a constant quantity), and increasing/decreasing exponentially (by a constant factor or percent). EXERCISE 3 1. Consider the data in the following tables: x y x y x y x y Which of them have linear behavior? (No need to find formulas yet!). Explain in terms of rates of change Consider the data sets with linear behavior. Pick a point and use the slope you found in part (2) to find three more point Find the algebraic expression to describe the data sets that are linear Use technology to plot the data Graph the mathematical model you found together with the data and see that the data points satisfy the mathematical model. Linear Functions. Revised 9/9/2010 Page 1

2 2. A car is 30 kilometers away from a city and start moving at a constant velocity of 50 km/hour Generate a table to determine the distance from the city during the first 4 hours, recording the information every half and hour ( t = 0.5) 2.2. Find a mathematical expression that represents the distance of the car from the city as a function of time, in hours Use the mathematical expression to determine: How far the car has traveled after 3.5 hours How far is the car from its starting point after 3.5 hours? How long it takes the car to be 135 km away from the city. Show work 3. We know that water freezing point is zero in the Celsius scale ( 0 0 C ) which corresponds to 32 in the Fahrenheit scale (32 0 F), and that the water boiling point is 100 in the Celsius scale ( 0 C) or 212 in the Fahrenheit scale ( 0 F) Write F (Fahrenheit) degrees as a linear function of C (Celsius) degrees 3.2. Write C degrees as a linear function of F degrees Interpret the meaning of the slope in parts 3.1 and 3.2. Include the units. Linear Functions. Revised 9/9/2010 Page 2

3 4. Two lines are perpendicular if the product of their slopes is Find the equation of the line passing through the points (-1, 2) and (3, 4.5) Find the equation of the line perpendicular to the line you found in 4.1 that goes through the point (0,3) 4.3. Find the points, which are 5 units from (-1, 2), and lie on a line perpendicular to the line in The point (3, 4) is on a line with slope Use the interpretation of the slope (do not find the equation of the line) to find three more points on the line Find the equation of the line and use it to find three more points on the line Find the equation that will characterize all the lines parallel to this line Use the expression from 5.2 to determine the equation of parallel passing through the point (3, -10). Linear Functions. Revised 9/9/2010 Page 3

4 6. Find the equation of the line perpendicular to the line and passing through the point (2,50). (HINT: You may want to sketch the lines first) 7. Consider the line passing through the points (2,5) and (2,0) Sketch the line and find its equation Find the equation of the line parallel to it and passing through the point (0,2) 7.3. Find the equation of the line perpendicular to it and passing through the point (0,2) Linear Functions. Revised 9/9/2010 Page 4

5 Exponential Growth. The table below gives the population of Mexico in millions (estimated) from 1980 to 1986 Year Population (millions) Once again, instead of using the year as input, we start with t=0, that corresponds to the initial time in our discussion, it is Thus, the table looks like: t (years) P=P(t) Population in millions t years after 1980 EXERCISE 1 1. From the table can you determine whether the function increases or decreases? 2. Enter the data in the calculator and see what the plot looks like. Does it look linear from the plot? 3. Use the data from the table to justify why P(t) is not a linear function. 4. Determine the concavity of this function. HINT: Look at the slope of the line segment of consecutive points. EXERCISE 2 Enter in the calculator the expression P( t) = 67.38( 1.026) t 1. Generate a table of values to see the population year by year starting at t=0. 2. From the table determine (an approximation is fine) how long it takes for the initial population to double. You can find this result more accurately by changing the values in the table. Call this value t d. t d = Linear Functions. Revised 9/9/2010 Page 5

6 3. Look at the population at t=5. To this number add the number t d from (2) and look at the population at that time. How is it compared to the population at t=5? Repeat the same process for t= Make a conjecture as to what happens to the population t d years later after any given time. 5. You can find t d using the graph of the exponential function. To do this, graph the exponential function and the horizontal line corresponding to twice the original population. The first coordinate of the intersection point will give you the time when the initial population doubles. It has to be t d. Find the exact doubling time for the population of Mexico. EXERCISE 4 1. A population can be modeled by the expression a. What is the initial population? b. What is the growth factor for this population? c. What is the rate of growth per year of this population? d. What is the rate of growth every four year? 2. The estimated population of the United States in July 2005 is 295,734,134. It is also estimated that the population grows at a rate of 0.95% per year. a. Write an expression to calculate the population of the United States as a function of time, in years, starting in July b. What would be the rate of growth per month? Linear Functions. Revised 9/9/2010 Page 6

7 c. What would be the doubling time of the USA population? Determine this value using tables and using the graph of the exponential function. Exponential decay EXERCISE 5 Consider the data given by the table below. It contains information about the amount of carbon-14 present after t years. t (years) C (amount present) From the data determine whether this function increases or decreases. From the table justify why the data does not have linear behavior but it has exponential behavior. Justify the concavity of this function. Plot the data. From the plot can you determine whether the data has linear or exponential behavior? Putting all the information together you should have that Because C(2)=C(1)* , C(3)=C(2)* now our exponential function is decreasing. That s why it is called exponential decay. The number indicates that % of the substance stays and 0.013% ( = ) decays. We have that 50 is the initial amount, the rate of decay is r=0.013%, and the decay factor is 1-r = EXERCISE6 1. Show a table of values for C(t), t=0,, Estimate the time required so that the amount present is half of the initial one. Call this number t h. 3. What is the amount present at t=2?. 4. What is the amount present at the time t+ t h? 5. Make a conjecture based on the results of (4) and (5). Verify your conjecture with another value. The number t h you found is called the half-life time of the substance. 6. In your own words state your interpretation of the half-life time of a substance. 7. You can also find the half-life graphically. To do this, graph the exponential function and the horizontal line corresponding to half the initial amount of the substance. The first coordinate of the intersection point will give you the time when the initial amount is reduced to half. It has to be t h. Linear Functions. Revised 9/9/2010 Page 7

8 The graph of this function is shown below. Some observations about this graph: o This function is defined for any real number. So its domain is (-, ) o The Y-intercept corresponds to the initial amount. o The function is decreasing and concave up o As the values of t become more negative (t-> - ), the values of C(t) gets more positive (C(t)-> ) o The graph never intercepts the X-axis but gets as close to it as one pleases. The X- axis is a horizontal asymptote o As the values of t become more positive (t-> ), C(t) is positive but closer to zero (C(t)-> 0 +, or C(t) approaches to zero from values larger than zero). SUMMARY The exponential function can be of the type exponential growth or exponential decay. The general equation of an exponential function is ( ) = P 0 a t P t where represents the value of the function when t=0, initial population or initial amount, and a represents the growth or decay factor. When it is exponential growth and when it is exponential decay. The graphs are like the ones we obtained above. If, a is the growth factor and r is the rate of growth, given as percent. If, a is the decay factor and r is the rate of decay, given as percent. For example, if a=1.35, the population has a growth factor of 1.35, which means it is growing at 3.5% On the other hand, if a=0.9, the population has a decay factor of 0.9, and the population is decay at the rate of 1-0.9=0.1 or 10%. From the graph and a table of values we infer that behavior: has the following Linear Functions. Revised 9/9/2010 Page 8

9 1. As t gets more negative, gets closer to zero but still positive. We represent this as 2. As t gets more positive gets more and more positive getting bigger than any given number we choose. This is represented as Likewise, from the graph and a table of values we infer that has the following behavior: 1. As t gets more negative, gets more and more positive getting bigger than any given number we choose. This is represented as 2. As t gets more positive is positive but gets closer to zero. We represent this as NOTE: There is a special number, which belongs to the family of the number, and appears often in exponential functions. It is the number. Since, the function will be concave up, increasing, and always positive. EXERCISE 7 1. Consider the two data tables below. Verify exponential function a. Verify that each one has exponential behavior. x y x y b. Find the growth or decay factor per period of time and per year. c. Find the rate of growth or decay per year. d. Determine the doubling/half-life time. e. Write two mathematical expressions to model each data. 2. Consider the points (-1, 7), (3, 0.5) a. Find a linear model that passes through these points. b. Find an exponential model that goes through those points. c. For the exponential model interpret the information provided by it. 3. Consider the function. Its graph looks very much like the graph of. Use a table of values for to convince yourself that this graph does not correspond to an exponential function. 4. The following sequences have exponential or linear behavior. Find an expression for the general term of each of them and graph them both as terms in the domain and as a function. a. Linear Functions. Revised 9/9/2010 Page 9

10 b. c. d. 5. You are going to cross a room by jumping certain distance each time. The distance from wall to wall is 10 meters, and each second you will jump half of the distance remaining between you and the wall you want to reach. a. Write a sequence that represents the length of each jump. b. Write a sequence that represents the remaining distance to cover. c. Which of these sequences represents an exponential sequence? d. Does the other sequence have a linear behavior? Explain. e. When will you reach the other wall? 6. $25,000 is invested at 3% every six months, in the Quick Pay Bank. a. Construct a table of values to see how much you have in the bank in the first 4 years. b. Estimate how long it takes to have $50,000 in the bank. c. Use graphs to find exactly how long it takes to have $50,000 in the bank. d. Use technology to verify your answer. Linear Functions. Revised 9/9/2010 Page 10