IB Math SL Year 2 Name: Date: 2-1: Laws of Exponents, Equations with Exponents, Exponential Function

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Jerome Vincent Fletcher
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1 Name: Date: 2-1: Laws of Exponents, Equations with Exponents, Exponential Function Key Notes What do I need to know? Notes to Self 1. Laws of Exponents Definitions for: o Exponent o Power o Base o Radical o Index o Root 2. Solving Equations with Exponents Calculation for/process: o Perform Log Rules o Change bases o Rewriting Exponents Definitions for: o Exponent o Power o Base o Radical o Index o Root Calculation for/process: o Perform Log Rules o Change bases o Rewriting Exponents o Expand exponential expressions 3. Exponential Functions Definitions for: o Exponential Functions o Graph Transformations Calculation for/process: o Expand exponential expressions o Graph Transformations In this lesson we will revisit the following learning goals: 1. What are the laws of exponents? 2. How do we solve equations where the variable is in the exponent? 3. What are the properties of exponential functions? 4. How can we sketch the graph of an exponential function? 5. What are the properties of e? What is e?

2 What is a 2-1 Loop? This practice has multiple loops, with parts in each loop. Part 1: people work together on a designated topic Part 2: person works on the same designated topic. Before moving on, the pair will and make corrections Laws of Exponents (and Expanding with Exponents) Exponent Rule In Words Example (by expanding) When with the base, keep the base the same and the exponents. = ( ) When with the base, keep the base the same and the exponents., keep the base the same ( ) = When an exponent is raised to a and the exponents. Any base raised to the power is 1. = = = ( ) m When a power and a root are involved, the part of the fractional exponent is the and the part is the. = Any base raised to a power can be rewritten as a fraction where the numerator is and the denominator is that same base raised to the power. ( ) = When a fraction is raised to an exponent, the exponent to each part of the fraction. ( ) Note:

Using the laws of exponents to solve exponential equations 1. Rewrite the problem, such that both sides of the equation have the. (This will require using laws of exponents.) For example, 2.

Exponential Functions An exponential function is a function of the form ( ), where is a positive real number,, and.

3 Using the laws of exponents to solve exponential equations 1. Rewrite the problem, such that both sides of the equation have the. (This will require using laws of exponents.) For example, 2. Since the bases are, we can drop the bases and set the equal to each other. 3. Solve algebraically. Exponential Functions An exponential function is a function of the form ( ), where is a positive real number,, and. o controls how steeply the graph increases or decreases o controls horizontal translation o controls vertical translation o is the equation of the horizontal asymptote *Note, you can always type the function in your GDC if you are unsure. We can sketch reasonably accurate graphs of exponential functions using: o The horizontal asymptote o The y-intercept o ONE other points, for example, when x = 2 The Natural Exponential Function: f(x) = e x The graph of the exponential function f(x) = e x is a graph of exponential grow and the graph of f(x) = e -x is of exponential decay.

4 2-1 Loop Partner Work Loop 1: Expanding Exponents *****without a calculator 1. Expand ( ) 2. Without your calculator evaluate a) b) ( ) c) 3. Simplify this expression: ( ) with this section. Once both partners are completed, check answers with one another then move on to the next loops. Do not progress without your partner. 4. Expand ( ) 5. Without your calculator evaluate a) ( ) b) c) 6. Simplify this expression: Once you checked above, try this with your partner ( )

5 Loop 2: Solving Equations with Exponents ****NO CALCULATOR 7. Solve for x: 8. Solve for x: 9. Solve for x: with this section. Once both partners are completed, check answers with one another than move on to the next loop. Do not progress without your partner. 10. Solve for x: 2 x = Solve for x: = Solve for x: 5 2x-1-25 = Solve for x: 3 1-2x =243

6 Loop 3: Sketching Exponential Curves **WITHOUT A CALCULATOR 14. Given the graph of f(x), sketch the graph of g(x) on the same set of axes showing clearly any intercepts on the axes and any asymptotes a) Consider: ( ) a) What is the y-intercept? b) What is the equation of the horizontal asymptote? c) What is f(-2)? d) Hence, sketch the graph. Remember. 1. What is the y-intercept? 2. What is the equation of the horizontal asymptote? 3. What is a point on the curve? 4. Hence, sketch the graph on the plane. with this section. Once both partners are completed, check answers with one another than move on to the next loop. Do not progress without your partner. 15. Given the graph of f(x), sketch the graph of g(x) on the same set of axes showing clearly any intercepts on the axes and any asymptotes Remember. 1. What is the y-intercept? 2. What is the equation of the horizontal asymptote? 3. What is a point on the curve? 4.Hence, sketch the graph on the plane.

7 Loop 4: Exponent Laws **WITHOUT A CALCULATOR 19. Simplify these expressions a) Show that: b) c) ( ) with this section. Once both partners are completed, check answers with one another than move on to the next loop. Do not progress without your partner. 20. Simplify these expressions a) b)

8 Loop 5: Solving Equations with Exponents ****NO CALCULATOR 21. Solve for x: 22. Solve for x a) 5 3x = 25 x-2 Work on this set of problems individually. DO NOT WORK WITH YOUR PARTNER for this portion. Once both partners are completed, check answers with one another than move on to the next loop. Do not progress without your partner. b) 9(3 3x+1 )=

9 Loop 6: Applications calculator allowed! 23. A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 2000, there were 280 taxis in the city. After n years the number of taxis, T, in the city is given by T = n. (i) Find the number of taxis in the city at the end of (ii) Find the year in which the number of taxis is double the number of taxis there were at the end of with this section. Once both partners are completed, check answers with the key. You have completed your practice today. 24. Show all work:

10 Loop 7: Sketching Exponential Curves **WITHOUT A CALCULATOR 16. The diagram shows the sketch of ( ). Sketch the graph of ( ) on the same set of axes. 17. Given the graph of f(x), sketch the graph of g(x) on the same set of axes showing clearly any intercepts on the axes and any asymptotes with this section. Once both partners are completed, check answers with one another than move on to the next loop. Do not progress without your partner. 17. Given the graph of f(x), sketch the graph of g(x) on the same set of axes showing clearly any intercepts on the axes and any asymptotes Write your 3 qualities required to sketch a graph here:

Name Homework Packet Week #12

Name Homework Packet Week #12 1. All problems with answers or work are examples. Lesson 4.4 Complete the table for each given sequence then graph each sequence on the coordinate plane. Term Number (n) Value of Term ( ) 1 2 3 4 5 6