76 Chapter 5 More About Exponents and Roots Lesson 5.5 Equations with Radicals 77 Lesson 5.5 Equations with Radicals Evaluate the formula to find the speed the cars Activity 1 Solving Radical Equations must reach on each roller coaster. Because the vertical loops used in roller coasters are not After solving a radical equation, remember to check for extraneous solutions. perfect circles, the total height of the loop is about 2.5 times its radius. 1. Solve Vx - 6 =2 Viper: Gash: 2. Solve VI - 3+5 =1 Shockwave: anolfor 2. You would like your roller coaster to have a vertical loop that is 200 feet tall. How fast must the cars travel? 3. Solve 3V/4x -1= -15 3. Now suppose you know the maximum speed possible for the cars on a particular roller coaster. Can you calculate the height of the tallest vertical loop the cars can negotiate? Here are the speeds of the world's three fastest roller coasters Activity 2 Roller Coasters You have been commissioned to design a new roller coaster for an Speed Roller Coaster Location 86 mph Fujiyama Fujikyu Highland Park, Japan amusement park. The roller coaster should be more spectacular than all existing Steel Phantom Kennywood, West Miflin, Pennsylvania roller coasters, and in particular it should include a vertical loop. In order to stay 80 mph 79 mph Desperado Buffalo Bill's, Jean, Nevada on the track through the loop, the cars must travel at a speed given (in miles per hour) by Calculate the maximum loop height possible for each roller coaster. v = V89.3r where r is the radius of the loop in feet. Here are the three roller coasters with the tallest vertical loops: Desperado: Height Roller Coaster Location 188 feet Viper 173 feet Gash Six Flags Magic Mountain, Valencia, California 170 feet Shockwave Six Flags Great Adventure, Jackson, New Jersey Six Flags Great Adventure, Gurnee, Illinois78 Chapter 5 More About Exponents and Roots Steel Phantom: Fujiyama: Activity 3 Quadratic Equations 1. Solve (2x + 1)2 =8goo sheva overt of isfeeds mellon mov patrolnow you 2. Solve 3x2 - 6x +2 =0, and simplify the solutions. Wrap-Up In this Lesson we practiced the following skills: Solving radical equations Checking for extraneous solutions Simplifying solutions of quadratic equations 1. What operation might introduce extraneous solutions? 2. What is wrong with this strategy: To solve an equation with a variable under a square root, square each term of the equation. 3. How many solutions does the equation x2 = 64 have? How many solutions does 23 = -64 have?32 Chapter 6 Functions 6. Look back at the table. What is happening to the number of new infected Lesson 6.1 Definitions and Notation residents each day as time goes on? How is this phenomenon reflected in the graph? How would your graph look if the number of new patients every day -Activity 3 Functions and Graphs were a constant? The table shows data obtained while heating a solid sample of stearic acid, a waxy solid used in making candles, soap, and some plastics. Heat was applied at a constant rate throughout the experiment. Time (min) 0 0.5 1.5 2 2.5 3 4 5 6 7 8 8.5 9 9.5 10 Temp (C) 19 29 40 48 53 55 55 55 55 55 55 64 70 73 74 7. Summarize your work: In your own words, describe how the number of - a. Plot the data with time on the horizontal residents infected with cholera changes with time. Include a description of axis and temperature on the vertical axis. Is 70 your graph. Is I a function of t? Why or why not? temperature a function of time? Why or why not? DID WOH b. Choose variables and write an equation using function notation. Label the axes on your graph. c. Is temperature a linear function of time? Why or why not? Describe the temperature - Activity 2 Function Notation as a function of time. Data indicate that U.S. women are delaying having children longer than their counterparts 50 years ago. The table shows P = f(t), the percent of 20-24 year old women in year t who had not yet had children. (Source: U.S. Dept of Health and Human Services) d. Energy (in the form of heat) is required to raise the temperature of a Year, t 1960 1965 1970 1975 1980 1985 1990 1995 2000 substance, and it is also needed to melt a solid substance to a liquid. By % of Women, P 47.5 51.4 47.0 62.5 66.2 67.7 68.3 65.5 66.0 analyzing the graph, what do you think is the melting point of stearic acid? How long did it take the sample to melt? X a. How does the table support the assumption that P is a function of t? Does this assumption make sense for the context of the problem? Wrap-Up In this Lesson, we worked on the following skills and goals related to functions: b. Evaluate f(1985) and explain what it means. Recognizing functions Evaluating functions defined by tables, graphs, or equations Using function notation c. Estimate a solution to the equation f(t) = 68 and explain what it means. 1. In Activity 1, what is the input variable, and what is the output variable? 2. In Activity 2, can you find a formula for P in terms of t? Does this mean that P is not a function of t? Why or why not? 3. In Activity 3, is time a function of temperature? Why or why not? 4. In Activity 3, how can you estimate the temperature of the sample at times d. In 1997, 64.9% of 20-24 year old women had not yet had children. Write an equation with function notation that states this fact. not listed in the table?86 Chapter 6 Functions Lesson 6.2 Graphs of Functions Lesson 6.2 Graphs of Functions 87 Activity 1 The Graph of a Function Activity 2 Functions Defined by Graphs The graph shows the U.S. unemployment rate as a function of time, U = f(t), for the years 1985-2004. (Source: U.S. Bureau of Labor Statistics) The figure shows the graph of the function y = F(s). A solid dot means that the point is part of the graph, and U.S. Unemployment Rate an open circle means that the point is not part of the graph. Rate (%) a. Find F(-3), F(-2), and F(2). 1985 1990 1995 2000 Year b. For what value(s) of s is F(s) = -1? a. Evaluate f (1993), and explain its meaning for this problem. c. Find the maximum value of F(s). For what value(s) of s does F take on its maximum value? b. When did the unemployment rate reach its highest value, and what was its highest value? Write your answer in function notation. d. Find the minimum value of F(s). For what value(s) of s does F take on its minimum value? c. When did the unemployment rate fall to its lowest value, and what was its Activity 3 Solving Equations Graphically lowest value? Write your answer in function notation. 1. Use the graph of 2. Use the graph of y = f ( ) = ( 2 - 2)2 B = f(p) = 2p3 -3p+2 to solve 3p3 - 3p + 2= 2 d. Solve the equation f(t) = 4.5, and explain its meaning for this problem. to solve (x -2)2 =1 e. Over what intervals was the unemployment rate increasing? - X