Unit 5 Exponential Functions. I know the laws of exponents and can apply them to simplify expressions that use powers with the same base.

Transcription

1 Unit 5 Exponential Functions Topic : Goal : powers I know the laws of exponents and can apply them to simplify expressions that use powers with the same base. 5.1 The Exponent Rules Multiplying Powers With the Same Base 3 4 x 3 6 RULE When you multiply two powers with the same base, you simply have to keep the same base and add the exponents. Dividing Powers With the Same Base RULE When you divide two powers with the same base, you simply have to keep the same base and subtract the exponents.

2 Powers of Powers (3 4 ) 6 RULE When you have a power of a power, you keep the base the same and multiply the exponents. Example 1. Use the Exponent Rules to simplify the following (3 6 ) 3 3 ( ) 3 5

3 Example 2. Use the Exponent Rules to simplify the following algebraic expressions Example 3. Exponents and probability Probability = # of outcomes desired total # of any outcome a) A multiply choice question has 5 choices for answers. If you guess at the answer without reading it, what are your chances of getting it correct? b) If you guess at 5 questions, what are your chances of getting them ALL correct? Homework Page 285 #1 10, 12, 14, 16

4 Topic : Goal : exponents I know what a negative exponent means and I can evaluate expressions that have a negative exponent. 6.2 Evaluate Powers with Integer Exponents Before we talk too much about negative exponents, please remember the exponent laws we learned last class. They still apply. Now we know what a power actually means... a repeated multiplication 5 3 means 3 5's multiplied together. 5 3 means what? We can't have 3, 5's. Let's follow a pattern = 2 2 = 2 1 = 2 0 = 2 1 = 2 2 = 2 3 = 3 3 = 3 2 = 3 1 = 3 0 = 3 1 = 3 2 = 3 3 = We can think of negative exponents as repeated division. Also notice that the value of the negative exponent is just the reciprocal of the corresponding positive exponent.

5 The general rules for integer exponents are a 0 =1 (as long as "a" is NOT ZERO) 2. a n = 1 a n 3. a b ( ) n = b ( ) n a Example 1. Evaluate the following. Leave your answer as a fraction. a) 2 5 b) 4 2 c) 5 3 Evaluate as if it were a positive exponent, then take the reciprocal. Example 2. Evaluate the following. 1 a) ( ) 3 5 b) 1 ( ) 5 2 c) 2 ( ) 2 3 Take the reciprocal of the fraction, then apply the exponent.

6 Example 3. Express as a power with a positive base. a) 1 32 b) 1 27 c) Example 4. Express each as an exponent with a base of 2. a) (4)3 b) (16) 6 c) (64) 3 Write the base as a power of 2, then apply the exponent law for a power of a power. Homework Page 293 #1 11, 13, 14, 16

7 Topic : Goal : exponents I know what a rational exponent means and how to evaluate it. 5.3 Rational Exponents A quick investigation... Also... How about... (if we reverse our power of a power law, we can break it down into two things we already know) All regular exponent rules apply. Example 1. Evaluate the following WITHOUT a calculator...

8 Example 2. Evaluate the following WITH a calculator...

9 Topic : Properties of Exponential Functions Goal : I can explain/identify the properties of exponential functions including; domain, range, growth or decay and rate of increase. 5.4 Properties of Exponential Functions Investigating the properties of Exponential Functions with graphs. Applications and the general equation

10 Example 1: The Town of Mitchell has 2500 people. The population is growing at 7% every 3 years. Answer the following: A) Determine an equation to model this situation. B) Determine the population after 10 years. 25 years. Example 2: The number of bacteria triples every 6 hours. If 5 bacteria were present on a plate jammed under Alex's bed at 6:00 pm, answer the following: A) Determine an equation to model this situation. B) Determine how bacteria are present in the morning at 7:00 am. C) Determine how many bacteria are present in 1 week when Alex finally takes the dish to the kitchen.

11 Topic : Linear vs. Quadratic vs. Exponential Functions Goal : I can explain/identify the differences between linear, quadratic and exponential rates of growth and their applications. 5.5 Comparing Linear, Quadratic and Exponential Functions Linear, Quadratic or Exponential from a Graph Linear, Quadratic or Exponential from an Equation Consider the relations: y = 2x + 1, y = x 2 1, y = 2 x

12 Linear, Quadratic or Exponential from a Table of Values Consider the relations: y = 2x + 1, y = x 2 1, y = 2 x

13 Lesson6.notebook January 08, 2013 Applications of Exponential Functions Today's goal: I can extend my knowledge of solving exponential equations to real world situations and reflect on the reasonableness of my answer. Financial Natural Sciences

14 Lesson6.notebook January 08, 2013 Mathematical Modeling Example: Barry has decided to invest $5000 in an account that earns 8% per annum compounded quarterly. Answer the following: A) Develop and equation to model Barry's investment. B) How much will Barry's investment be worth in 10 years. C) How long will it take Barry's investment to triple?

15 Lesson6.notebook January 08, 2013 Example: Greg has a pet ant farm. He started the colony with 25 ants and has measured the growth rate to be 18% in population per year. Answer the following: A) Determine an equation to model this situation. B) Determine how many ants he will have in 5 years. C) Determine how many years it will take to have 1000 ants. Example: Owen has "borrowed" 150 g of radioactive material from a nearby nuclear facility and has stashed it in his closet. If the material has a half life of 80 days, answer the following: A) Determine a model for this situation. B) Determine how much material is left after 120 days. C) Determine how long it will take for their to be 1 g of material left.

16 Lesson6.notebook January 08, 2013 Example: Alycia has invested $2500 for 10 years. If the investment was compounded monthly and is now worth $4200, determine the interest rate at which it was invested. Homework: pg 19, All questions