Section 3.4 Square Root Functions The key to identifying the equation of a square root function is that the independent variable is under the radical. Which functions are square root functions? g( x) x 5 h( x) x 5 f ( x) x 5 Enter f ( x) x in your y= menu. Examine the table, beginning with x = -2. Complete the following table and sketch the graph. 4 9 Exact f(x) f(x) (real no.) Identify the domain and range. Use the table feature of your calculator to help you sketch each of the following functions. Remember the values given in the table are approximate real numbers. Give the domain and range in interval notation. g( x) x h( x) x 3 f ( x) x 3 Winter 2015 1
f ( x) x 1 g( x) x 1 h( x) 3 x Applications The velocity of a popular sedan is related to its stopping distance by the equation, v 3.95 d, where d is the stopping distance in feet and v is the velocity (in mph). (a) If the sedan stopped in 70 feet, how fast was it traveling? (b) Sketch a graph of the velocity function. (c) Estimate from the graph how many feet it would take the sedan to stop if it was traveling 30 mph. Section 3.5 Exponential Functions An exponential function is a function that can be represented by the equation f ( x) a( b) x for b > 0 and b 1 where a is the vertical intercept (0, a) and the base, b, is the common ratio. The key to recognizing the equation of an exponential function is that the independent variable is the exponent. Winter 2015 2
For the exponential function f ( x) a( b) x, and a > 0 If b > 1, then f (x) is an exponential growth function. If 0 < b < 1, then f (x) is an exponential decay function. Identify the exponential functions in the following list. For those that are exponential, identify a and b and tell whether they represent growth or decay. (a) y 3 x (b) m 4(2) x (c) f ( x) 1 4 3 x (d) g x x 3 5 4 Complete the following table and graph the following exponential function. f x 2 x Complete the following table and graph the following exponential function. f x x 1 3 A horizontal is a horizontal line that a graph gets closer and closer to, but never reaches or intersects with it. The range of an exponential function is related to the horizontal asymptote. Both of the functions we graphed have a horizontal asymptote at the horizontal line. Winter 2015 3
Identifying Exponential Functions from a Table A function is said to be an exponential function if equal steps in the independent variable produce equal ratios for the dependent variable. f x 3 x 1 1 1 3 9 27 9 3 Determine whether the table represents and exponential function. If it is exponential, find the equation of the function and tell whether it is an exponential growth function or exponential decay function. g (x) 75 15 3 3 5 3 25 3 125 n -1 0 1 2 f(n) -2 1 4 7 t -2-1 0 1 2 h(t) 5 4 5 2 5 10 20 Winter 2015 4
Section 3.6: Applications of Exponential Functions Review: An exponential function is a function that can be represented by the equation f ( x) a( b) x for b>0 and b 1 where a is the vertical intercept and the base, b, is the common ratio. Find the equation of the exponential function for each table. Identify whether the equation represents a growth or decay function. t 0 1 2 3 4 H(t) 0.8 2.4 7.2 21.6 64.8 x -1 0 1 2 3 F(x) 80 8 0.8 0.08 0.008 Modeling Growth and Decay A bacteria colony began with 100 bacteria. Its population is modeled by Pt ( ) 100(1.09) t. Where t is the number of elapsed hours and P(t) is the population at time t. Predict the population for 12 hours? 24 hours? 72 hours? Finding the Model A bacteria colony began with 1000 bacteria. After 1 hour, there were 1200 bacteria. After 2 hours, there were 1440 bacteria. Find the exponential equation which models this situation and use the equation to predict the number of bacteria at 60 hours. Winter 2015 5
The number of women in the labor force in the United States has been steadily increasing since 1890 when there were 3.7 million women working. The rate of yearly increase is 3.01% (Source U.S. Bureau of Census). 1. Use t = 0 to represent the year 1890 and write and exponential model for the growth in the number of women in the labor force. 2. Use your model to estimate the number of working women in 1950. 3. Use your model to predict the number of women who will be working in 2010. Examine Example 3 on page 89 and Practice 3 on page 91 Use your calculator to make a scatter plot (see pages 83 85 Module 2) of the data, find both linear and exponential models of the data, and then use the appropriate model (the equation whose correlation value closest to 1 or -1) to answer the questions. x 1 3 5 6 8 12 14 y 2 6 11 14 19 28 33 Linear model (r = ) Exponential model (r = ) Find the value of y when x is 10. Find the approximate value of x when y is 98. s 1 4 6 9 10 t 62 125 192 375 468 Linear model (r = ) Exponential model (r = ) When s is 15, what is the value of t? Winter 2015 6