10.2 Graphing Exponential Functions
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1 Name Class Date 10. Graphing Eponential Functions Essential Question: How do ou graph an eponential function of the form f () = ab? Resource Locker Eplore Eploring Graphs of Eponential Functions Eponential functions follow the general shape = ab. Graph the eponential functions on a graphing calculator, and match the graph to the correct function rule. 1. = 3 (). = 0.5 () 3. = 3 (0.5) 4. = -3() a. b. c. d. Houghton Mifflin Harcourt Publishing Compan B In all the functions 1 4 above, the base b > 0. Use the graphs to make a conjecture: State the domain and range of = ab if a > 0. In all the functions 1 4 above, the base b > 0. Use the graphs to make a conjecture: State the domain and range of = ab if a < 0. What is the -intercept of ƒ () = 0.5 ()? Module Lesson
2 Note the similarities between the -intercept and a. What is their relationship? Reflect 1. Discussion What is the domain for an eponential function = ab?. Discussion Describe the values of b for all functions = ab. Eplain 1 Graphing Increasing Positive Eponential Functions The smbol represents infinit. We can describe the end behavior of a function b describing what happens to the function values as approaches positive infinit ( ) and as approaches negative infinit ( - ). Eample 1 Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. ƒ () = Choose several values of and generate ordered pairs. f () = a = 1 b = -intercept: (0, 1) End Behavior: As -values approach positive infinit ( ), -values approach positive infinit ( ). As -values approach negative infinit ( - ), -values approach zero ( 0). Using smbols onl, we sa: As,, and as -, 0. (-1, 0.5) (1, ) (0, 1) f() = (, 4) 0 4 Houghton Mifflin Harcourt Publishing Compan Module Lesson
3 B ƒ () = 3 (4) Choose several values of and generate ordered pairs. f () = 3 (4) a = b = -intercept: (, ) End Behavior: As, and as -,. Reflect 3. If a > 0 and b > 1, what is the end behavior of the graph? Describe the -intercept of the eponential function ƒ () = a b in terms of a and b. Your Turn 5. Graph the eponential function ƒ () = () After graphing, identif a and b, the -intercept, and the end behavior of the graph. Houghton Mifflin Harcourt Publishing Compan f () = () Module Lesson
4 Eplain Graphing Decreasing Negative Eponential Functions Eample Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. ƒ () = - (3) Choose several values of and generate ordered pairs. f () = -(3) a = - b = 3 -intercept: (0, -) End Behavior: As, - and as -, (0, -) (1, -) f() = -(3) B ƒ () = -3 (4) -18 (, -18) Choose several values of and generate ordered pairs. f () = -3(4) a = b = -intercept: (, ) End Behavior: As, and as -, Houghton Mifflin Harcourt Publishing Compan Module Lesson
5 Reflect. If a < 0 and b > 1, what is the end behavior of the graph? Your Turn 7. Graph the eponential function. ƒ () = -3(3) After graphing, identif a and b, the -intercept, and the end behavior of the graph. f () = -3(3) Eplain 3 Graphing Decreasing Positive Eponential Functions Eample 3 Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. Use inequalities to discuss the behavior of the graph. ƒ () = (0.5) Houghton Mifflin Harcourt Publishing Compan Choose several values of and generate ordered pairs. f () = (0.5) a = 1 b = 0.5 -intercept: (0, 1) End Behavior: As, 0 and as -,. f() = (-1, ) (1, 0.5) (0, 1) (, 0.5) Module Lesson
6 B ƒ () = (0.4) Choose several values of and generate ordered pairs. f () = (0.4) a = b = -intercept: (, ) End Behavior: As, and as -,. Reflect 8. If a > 0 and 0 < b < 1, what is the end behavior of the graph? Your Turn 9. Graph the eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. ƒ () = 3 (0.5) f () = 3 (0.5) Houghton Mifflin Harcourt Publishing Compan Module Lesson
7 Eplain 4 Graphing Increasing Negative Eponential Functions Eample 4 Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. ƒ () = -0.5 Choose several values of and generate ordered pairs. f () = a = -1 b = 0.5 -intercept: (0, -1) End Behavior: As, 0 and as -, -. B ƒ () = -3(0.4) Choose several values of and generate ordered pairs f() = -0.5 (-1, -) (, -0.5) 1 (1, -0.5) (0, -1) f () = -3(0.4) Houghton Mifflin Harcourt Publishing Compan a = b = -intercept: (, ) End Behavior: As, and as -, Module Lesson
8 Reflect 10. If a < 0 and 0 < b < 1, what is the end behavior of the graph? Your Turn 11. Graph the eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. ƒ () = - (0.5) f () = - (0.5) Elaborate 1. Wh is ƒ () = 3 (-0.5) not an eponential function? 13. Essential Question Check-In When an eponential function of the form ƒ () = a b is graphed, what does a represent? Houghton Mifflin Harcourt Publishing Compan Module Lesson
9 Evaluate: Homework and Practice State a, b, and the -intercept then graph the function on a graphing calculator. 1. ƒ () = (3). ƒ () = -() Online Homework Hints and Help Etra Practice 3. ƒ () = -5(0.5) 4. ƒ () = 3 (0.8) 5. ƒ () = (3). ƒ () = -4(0.) Houghton Mifflin Harcourt Publishing Compan Module Lesson
10 7. ƒ () = 7 (0.9) 8. ƒ () = -3 () State a, b, and the -intercept then graph the function and describe the end behavior of the graphs. 9. ƒ () = 3 (3) 10. ƒ () = 5 (0.) f () = 3 (3) f () = 5 (0.) Houghton Mifflin Harcourt Publishing Compan Module Lesson
11 11. ƒ () = - (0.7) 1. ƒ () = -4 (3) f () = - (0.7) f () = -4 (3) ƒ () = 5 () 14. ƒ () = - (0.8) f () = 5 () f () = - (0.8) Houghton Mifflin Harcourt Publishing Compan Module Lesson
12 15. ƒ () = 9 (3) 1. ƒ () = -5 () f () = 9 (3) f () = -5 () ƒ () = 7 (0.4) 18. ƒ () = () f () = 7 (0.4) f () = () Houghton Mifflin Harcourt Publishing Compan Module Lesson
13 19. Identif the domain and range of each function. Make sure to provide these answers using inequalities. a. ƒ () = 3 () b. ƒ () = 7 (0.4) c. ƒ () = - (0.) d. ƒ () = -3 (4) e. ƒ () = () 0. Statistics In 000, the population of Massachusetts was.3 million people and was growing at a rate of about 0.3% per ear. At this growth rate, the function ƒ () =.3 (1.003) gives the population, in millions ears after 000. Using this model, find the ear when the population reaches 7 million people. Houghton Mifflin Harcourt Publishing Compan Image Credits: Spirit of America/Shutterstock 1. Phsics A ball is rolling down a slope and continuousl picks up speed. Suppose the function ƒ () = 1. (1.11) describes the speed of the ball in inches per minute. How fast will the ball be rolling in 0 minutes? Round the answer to the nearest whole number. H.O.T. Focus on Higher Order Thinking. Draw Conclusions Assume that the domain of the function ƒ () = 3 () is the set of all real numbers. What is the range of the function? 3. What If? If b = 1 in an eponential function, what will the graph of the function look like? 4. Critical Thinking Using the graph of an eponential function, how can b be found? Module Lesson
14 5. Critical Thinking Use the table to write the equation for the eponential function. f () -1 4_ Lesson Performance Task A pumpkin is being grown for a contest at the state fair. Its growth can be modeled b the equation P = 5 (1.5) n, where P is the weight of the pumpkin in pounds and n is the number of weeks the pumpkin has been growing. B what percentage does the pumpkin grow ever week? After how man weeks will the pumpkin be 80 pounds? After the pumpkin grows to 80 pounds, it grows more slowl. From then on, its growth can be modeled b P = 5 (1.3) n, where n is the number of weeks since the pumpkin reached 80 pounds. Estimate when the pumpkin will reach 150 pounds. Houghton Mifflin Harcourt Publishing Compan Module Lesson
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LESSON Relating Slope and -intercept to Linear Equations UNDERSTAND The slope of a line is the ratio of the line s vertical change, called the rise, to its horizontal change, called the run. You can find
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COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural
21.1 Solving Equations by Factoring
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Name Date. Work with a partner. Each graph shown is a transformation of the parent function
3. Transformations of Eponential and Logarithmic Functions For use with Eploration 3. Essential Question How can ou transform the graphs of eponential and logarithmic functions? 1 EXPLORATION: Identifing
Derivatives 2: The Derivative at a Point
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Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
Solving Systems of Linear Equations by Graphing. ESSENTIAL QUESTION How can you solve a system of equations by graphing? 8.9 Slope-intercept form
? LESSN. Solving Sstems of Linear Equations b Graphing ESSENTIAL QUESTIN How can ou solve a sstem of equations b graphing? Epressions, equations, and relationships.9 Identif and verif the values of and
Graphs of Non-Linear Functions
Classwork Exploratory Challenge 1. Plot a graphical representation of the distance of the ball down a ramp over time. https://www.youtube.com/watch?v=zinszqvhaok Discussion 2. Did everyone s graph have
1Write and graph. 2Solve problems. Now. Then. Why? New Vocabulary
Direct Variation Then You found rates of change of linear functions. (Lesson -) Now Write and graph direct variation equations. Solve problems involving direct variation. Wh? Bianca is saving her mone
20.3 Applying the Zero Product Property to Solve Equations
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CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain
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Proportional and Nonproportional Situations
L E S S N. Florida Standards The student is epected to: Functions.F.1. Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal
Name Date Class Period. pencil straightedge graph paper How can you relate slope, y-intercept, and an equation?
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