Math Unit 3 Day 1-13 Notes

Name __________________________________ Unit 3 Day 1 Properties of Exponents Laws of Exponents: 1. ∙ = __________________ 2. ( ) = ___________________ 3. ( ) = ___________________ 4. 5. = _____________________ = ____________________ Examples: Where you __________ with more is where you should _______ with more! Simplify means → 1. Each base appears only once 2. No negative exponents 3. No power of powers Examples: 5 4 24 32 2 More Laws: 6. 7. 2 = ______________________ = ______________________ Recall: 0 = _________________ Examples: (3 ) ( ) 10 ∙ 10 2 A Negative exponent MUST be ______________________________ Negative exponents are _______________________________, move them to the ___________________________ of a fraction to make them happy!!!! Examples: (− 3) (5 ) Properties of Exponents Worksheet Evaluate each expression. No negative exponents and No decimals! 1. −5 2. (−5) 3. 6. 3 5. 4 7. 4. −6 8. Write the exponential expression in simplest form. No negative exponents and No decimals! )( ) )(−3 9. 2 11.( 12.(−2 10. 13. 17.( ) ( 14. 15. 18. 19. ) 16.(−3 ( ( ) ) 20. )(−3 ) ) Name __________________________________ Unit 3 Day 2 Rational Exponents Properties: 1. = ____________________ = _____________________ 2. Examples: 16 25 64 32 9 Properties of Exponents Worksheet Simplify the following completely. 1. 4 6. 32 ⁄ 11. 4 ⁄ 15. 2. 3. 7. ⁄ (81 19. 2 −16 2 ) ⁄ ⁄ ⁄ ⁄ ⁄ 3 ⁄ 4. 12. 6 (27 ⁄ ⁄ −3 ) 125 ⁄ 5. 9. 8. 16. ⁄ −64 ⁄ ⁄ 13. −3 ⁄ ⁄ 10. 4 ⁄ 14. ⁄ 17. ⁄ 20. −5 ⁄ 18. ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ 9 ⁄ ⁄ −4 ⁄ Name __________________________________ Unit 3 Day 3 Simplifying Radicals Square Root: Examples: √9 √24 −√48 √− 64 √54 Cube root: √ Recall the perfect cubes 1-7: Examples: √8 Pod People Story: Examples: √ √10 √6 Divide power by root Answer is new power outside 512 2 √40 − 3√135 2√2 Simplifying Radicals Worksheet – Simplify each radical expression. 1. √45 2. √75 5. √−250 6. 9. 2√32 − 3√98 24 3. √24 7. 4. √135 18 10.4√ 8. − √ 16 + 4 √8 Name __________________________________ Unit 3 Day 4 Add/Subtract/Powers of i = − 1 has no REAL solution, must use Imaginary numbers! Definition:√− 1 = Powers of : = _______, = _______, = _______, = ________, = ________, = _________, Remember the first 4 since the rest repeat! = _______,..... = ________, = ________ Examples: A. D. √− 50 C. √− 18 B. Complex Numbers: real and imaginary parts combined → (+)/(-): Examples: A. (3 + 4 ) + (7 + 10 ) B. (3 + 4 ) − (7 + 10 ) Powers/Operations of i Worksheet Evaluate the power of . 2. − 1. Write the complex number in standard form. 4. √−81 5. √−98 3. 6. 5 + √−49 7. 8 − √−18 Simplify and write the complex number in standard form. 8. (5 + 2 ) + (6 − 7 ) 9. (−2 − 4 ) − (5 − 8 ) 10. (1 − 3 ) + (7 − 2 ) 11. (−3 − 5 ) − (7 − 5 ) 13. 3 − (4 − 5 ) 12. 8 − (2 − 8 ) Name __________________________________ Unit 3 Day 5 Multiply/Divide i (x): Examples: A. √− 3∙ √− 5 B. (3 + 4 )(7 + 10 ) C. (3 + 4 ) D. (3 + 4 )(3 − 4 ) (÷): Complex Conjugate: 3 + 4 → ____________ Examples: A. B. Factoring with i: Examples: A. +4 B. C. + 25 C. + 81 Operations of i Worksheet Simplify and write the complex number in standard form. 1. 5 ∙ 8 3. (4 + 2 )(3 + 4 ) 2. √−50 ∙ √−2 4. 3(2 + 5 )(3 − 2 ) 5. (6 + 5 )(2 − 5 ) 6. 3 + √−4 2 − √−9 7. (3 − 5 ) 8. 9. 10. 11. 12. Factor the following Addition of Squares. + 16 14. + 49 13. 15. + 36 16. +9 Name __________________________________ Unit 3 Day 6 Rational Exponents and Quadratic Form Equations = __________________ Numerator = _______________ Denominator = ________________ To Solve: 1. Get exponent portion alone. 2. Cancel exponent by raising to the reciprocal power. 3. Watch for two answers when taking an even root 4. ALWAYS CHECK!!!!! Recall: Examples: Solve the following. = 25 A) C) 2 B) 4 − 47 = 115 = 108 D) 5 Quadratic Form: Factor and solve as if just Examples: E) + 7 − 18 = 0 + 4 = 44 . Watch out for ± answers. F) 4 − 25 Rational Exponent and Quad Form Worksheet Solve each equation containing a rational exponent on the variable. 1. ⁄ = 2 2. ⁄ = 5 3. ⁄ = 9 5. ⁄ = 27 6. ⁄ = 125 7. 3 Find all real solutions of each equation by factoring. 9. − 9 + 14 = 0 10. ⁄ 4. − 16 = 59 − 10 + 36 = 0 +9=0 8. 4 ⁄ = 81 ⁄ − 31 = 77 Name __________________________________ Unit 3 Day 7 Multiply/Divide Polynomials Rational Expression: Domain of a Rational Expression: Simplify a Rational Expression: Multiply: Divide: Examples: A. B. ∙ ÷ C. Multiply and Divide Polynomials Worksheet Simplify each rational expression. 1. 5. 8. 3. 2. ∙ ∙ 6. ÷ 9. ÷ 4. 7. 10. ∙ ÷ Name __________________________________ Unit 3 Day 8 Add/Subtract Polynomials + Recall: Adding or subtracting fractions… Add/Subtract: Examples: + A. B. − Add and Subtract Polynomials Worksheet Simplify each rational expression. 1. + 2. 4. − 5. 7. + 8. + + − 3. − 6. + 9. − Name __________________________________ Unit 3 Day 9 Polynomial Story Problems Examples: Solve the polynomial story problems. A. The number of singles tennis matches that can be played among n tennis players is given by the polynomial − . Find the number of singles tennis matches that can be played among four tennis players. B. On an expressway, the recommended safe distance between cars in feet is given by 0.015 + + 10, where v is the speed of the car in miles per hour. Find the safe distance when: 1. = 30 ℎ 2. = 55 ℎ Story Problems Worksheet Solve each story problem. 1. The air resistance (in pounds) on a cyclist riding a bicycle in an upright position can be given by 0.016 where v is the speed of the cyclist in miles per hour (mph). Find the air resistance on a cyclist when: A. = 10 ℎ B. = 15 ℎ 2. The volume of a right circular cylinder is given by height of the cylinder. Find the volume when: A. = 3 ℎ=8 , ℎ, where r is the radius of the base and h is the B. =5 ℎ = 12 3. Based on data from one experiment, the reaction time, in hundredths of a second, of a person to a visual stimulus varies according to age and is given by the expression 0.005 − 0.32 + 12, where x is the age of the person. Find the reaction time to the stimulus for a person who is: B. = 50 A. = 20 4. Find the number of chess matches that can be played among the members of a group of 150 people. Use the formula − . Name __________________________________ Unit 3 Day 10 Define and Evaluate Functions Function: x-values _______________________ in order to have a function. There are 3 ways to write a function: Set of Graph: Function if: Ordered Pairs: Equation: Function if: Function if: Examples: Determine whether each is a function. A) {(1,5), (3,2), (5,2)} B) {(1,5), (3,2), (3,4)} C) {( , ): = | |} (Recall: this is set notation…just look at the equation part!!!) Recall Function Notation: Steps: 1. 2. 3. ( ), ( ), ℎ( ) These are just ‘names’ of equations!!!!! Examples: A. Find (− 3) when ( ) = − 2 + 9 B. Find (3) when ( ) = 3 C. Find ( + ℎ) when ( ) = 4 − 2 D. Find (− + 2) when ( ) = 5 − 4 +4 Evaluating Functions Worksheet State whether the set of ordered pairs is a function. 1. {(2, 3), (5, 1), (−4, 3), (7, 11)} 2. {(5, 10), (3, −2), (4, 7), (5, 8)} 3. {(4, 4), (6, 1), (5, −3), (4, 5)} Evaluate each function. 4. Given ( ) = 3 − 1, find: a. (2) b. (−1) c. (0) d. e. ( ) f. ( + 2) b. (−1) c. (0) e. ( ) f. ( + 5) b. (2) c. (−2) 5. Given ( ) = 2 a. (3) + 3, find: d. 6. Given ( ) = √ a. (0) 7. Given ( ) = 3 a. (−4) 8. Given ( ) = 5 a. (−3) + 5, find: − , find: b. (− ) b. (0) c. ( + 1) c. ( + ℎ) Name __________________________________ Unit 3 Day 11 Composition of Functions Recall Function Notation: ( ), ( ), ℎ( ) These are just ‘names’ of equations!!!!! Sum: The sum of f and g, ( + )( ) = __________ + _________ Difference: The difference of f and g, ( − )( ) = __________ − _________ Product: The product of f and g, ( ∙ )( ) = __________ ∙ _________ Quotient: The division of f and g, ( ) = Domain: Remember that the denominator CANNOT = 0 so you need to check your fraction for excluded values, AKA the Domain. (answers that make the denom = 0) How? Examples: Find ( + )( ), ( − )( ), ( ∙ )( ) and , ( ) of each of the following. State the domain as well. A. ( ) = − + 2, ( ) = 4 + 3 ( + )( ) ( − )( ) ( ∙ )( ) ( ), Domain: What if you aren’t plugging in x??? Put the # in for every x you see but NEVER change the left side of the equation!!! Examples: B. Find ( + )(− 3), ( − )(− 3),( ∙ )(− 3)and (− 3) when ( ) = 5 − 4, ( )= +3 − 2 Composition: The composition of f and g means ( ( ))or ( ∘ )( ). Start from the __________ and work your way ________!!! Examples: A. Find ( (1)) and ( (1)) when ( ) = 2 + 1, ( ) = 3 + 4 B. Find ( ∘ )( ) when ( ) = 5 − 4 C. Find ( ( )) and ( ( )) when ( ) = 4 − 2, ( ) = +1 D. Find ( ∘ ∘ ℎ)( ) when ( ) = 3 , ( ) = − 4, and ℎ( ) = Composition of Functions Worksheet 1. Find a. ( + )( ), b. ( − )( ), c. ( ∙ )( ) and ,d. State the domain as well. ( )= − 2 − 15, ( ) = a. b. ( ) of each of the following. +3 c. d. Domain: 2. Find a. ( + )(−1), b. ( − )(−1), c. ( ∙ )(−1) and ,d. ( )= a. ( )= 2 −4 − 3 + 2, b. (−1) of each of the following. c. 3. Find a. ( ( )) and b. a. ( ( )) when ( ) = 3 + 5, b. 4. Find a. ( (2)) and b. a. ( (2)) when ( ) = d. Domain: ( )=2 −7 , ( )= 3 −5 b. Name __________________________________ Unit 3 Day 12 Inverses of Functions Recall: To be a function, x values ____________ repeat!!!! ( ) = and ( ) = . Inverse Functions: Functions are inverses if Notation: this is NOT an exponent!!! Graphs: mirror images!!! A function will have an inverse if the graph passes the ___________________ ______________________________. To Find: Flip the x and y values! Examples: Find the inverse of the following and tell if the inverse is a function. A set of ordered pairs: (3,5), (4,6), (9,10), (7,6), (2,5) Inverse: ______________________________________ Function: Yes No  Graph (Graph the inverse!) Function: Yes No Steps to find the inverse of an equation: 1. Replace ( ) with y. 2. Switch the x and y. 3. Solve for y. 4. Replace y with ( ). Examples: Find the inverse of each function. A. ( ) = 2 − 4 B. ( ) =______________ ( ) = ( − 3) ( ) =______________ Examples: Prove that the following are inverses of each other by using composition of ( ) = and ( ) = . functions. Remember: E. ( ) = + 3, ( ) = − 3 F. ( ) = 3 + 1, ( ) = Inverse Functions Worksheet Draw the inverse of the given graph. Is the inverse a function? 1. 2. Find the inverse of the function. 3. {(0, 1), (1, 2), (2, 4), (3, 8)} Inverse a function: circle one Inverse a function: circle one Inverse a function: circle one YES NO YES NO YES NO Use composition of functions to determine if the functions are inverses of each other. 5. ( ) = 3 , ( ) = 4. ( ) = , ( )= +3 ( ( )) Find the inverse of each function. 6. ( ) = 3 − 7 7. ( )= ( ( )) ( ( )) ( )= ( )= ( ( )) 8. ( )= ( )= +1