Section 5.5 Complex Numbers

Transcription

1 Name: Period: Section 5.5 Comple Numbers Objective(s): Perform operations with comple numbers. Essential Question: Tell whether the statement is true or false, and justify your answer. Every comple number is an imaginary number. Homework: Assignment 5.5. #1 1 in the homework packet. Notes: Vocabulary The conjugate of a b is a b. A comple number is a number that can be written in the form a + bi, where a and b are real numbers. Properties of Square Roots a 0, b 0 Product Property: a b a b Quotient Property: Simplifying Square Roots a b a b A square root is simplified if the radicand has no perfect square factor (other than 1) and there is no radical in the denominator of a fraction. E: Simplify the square root 7. Method 1 Step One: Find the largest perfect square that is a factor of Step Two: Rewrite 7 as a product using 36 as a factor. 36 Step Three: Rewrite as the product of two radicals. 36 Step Four: Evaluate the square root of the perfect square. 6 Method Step One: Rewrite 7 as a product of prime factors. 3 3 Reflection: 1

2 3 3 Step Two: Find the square root of each pair of factors Simplify the square root. Eample 1: 8 Eample : 175 Eample 3: 108 Eample 4: 147 Rationalizing the Denominator E: Simplify the epression 3 8. We must rationalize the denominator by multiplying by Now simplify the radical and the fraction Rationalize the denominator. Reflection:

3 Eample 5: 3 Eample 6: Imaginary Unit: i 1 Write in terms of i. Eample 7: 4 Eample 8: 64 Eample 9: 168 Eample 10: 64 Sum and Difference of Comple Numbers: Add or subtract the real parts and the imaginary parts separately. E: Find the sum: 3 5i 6 i E: Find the difference: 4 8i 3 i 3 6 5i i 3 7i 4 3 8i i 1 10i Perform the indicated operation. Write the result in the form a + bi. Eample 11: (5 + 4i) (-6 + i) Eample 1: (6 5i) + (-6 + 5i) Reflection: 3

4 Product of Comple Numbers: Use the distributive property or the bo method to multiply two comple numbers. E: Find the product 3 6i 5 i i i 15 6i 30i 1i 15 4i i Perform the indicated operation. Write the result in the form a + bi. Eample 13: (7i)(-14i) Eample 14: (3 + i) Eample 15: (4 6i)(4 + 6i) Eample 16: (15 5i)(3 + i) Comple Conjugate: The comple conjugate of a bi is a bi. Reflection: 4

5 Quotient of Comple Numbers: To divide two comple numbers, multiply the numerator and denominator by the comple conjugate of the divisor (denominator). E: Find the quotient 3 5 i 1 i. 3 5i 1 i 1 i 1 i 3 6i 5i 10i 1 4i 3 11i i i Note: The final answer is written in standard form. Perform the indicated operation. Write the result in the form a + bi. Eample 17: 6 7 3i Eample 18: 6 5i 9 4i Eample 19: 4i 3 5i Eample 0: 5 5i 7 5i Reflection: 5

6 Section 5.6 Square Root Property Objective(s): Solve quadratic equations by finding square roots. Essential Question: When is it necessary to simplify square roots? Homework: Assignment 5.6. # 39 in the homework packet. Notes: Solving a Quadratic Equation by Finding Square Roots E: Solve the equation Step One: Isolate the squared epression. Step Two: Find the square root of both sides Step Three: Solve for the variable. 6 or 6 Use the square root property to solve the equation. Eample 1: = 11 Eample : = 75 Eample 3: 7 = 0 Eample 4: + 36 = 0 Reflection: 6

7 Eample 5: 3 39 = 0 Eample 6: 14 = -4 E: Solve the equation n Step One: Isolate the squared epression. n 5 81 Step Two: Find the square root of both sides. Step Three: Solve for the variable. n 5 81 n 5 9 n 5 9 n 14 n 4 n 7 n Use the square root property to solve the equation. Eample 7: ( 3) = 5 Eample 8: ( + 4) = 17 Eample 9: ( + 3) = 5 Eample 10: (3 + ) = 10 Reflection: 7

8 Real-Life Application: Free Fall On Earth, the equation for the height (h) of an object for t seconds after it is dropped can be modeled by the function h 16t h 0, where h0 is the initial height of the object. E: A ball is dropped from a height of 81 ft. How long will it take for the ball to hit the ground? Use the free-fall function. h 16t h 0 h0 81, h 0 Initial height is 81 ft. The ball will hit the ground when its height is 0 ft. Solve for t. 0 16t 81 16t t 16 t 9 9, 4 4 Solution: Since time is positive, the only feasible answer is seconds Eample 11: ground? A ball is dropped from a height of 64 ft. How long will it take for the ball to hit the Eample 1: A penny is dropped from the top of the Stratosphere, a height of 1,149 ft. How long will it take for the penny to hit the ground? (Round to the nearest hundredth.) Reflection: 8

9 Section 5.7 Completing the Square Objective(s): Solve quadratic equations by completing the square. Essential Question: Homework: Assignment 5.7. #40 56 in the homework packet. Notes: Vocabulary The process of writing a quadratic equation so that one side is a perfect square trinomial is called completing the square. A quadratic equation is one that can be written in the form a + b + c = 0 where a, b, and c are real numbers and a is not 0. Use the bo method to multiply. Eample 1: ( + ) Eample : ( 4) Fill in the missing pieces. Eample 3: 0 + Eample 4: ( ) ( ) Reflection: 9

10 Eample 5: Eample 6: + + ( ) ( ) What value of c makes the left side of the equation a perfect square trinomial? Eample 7: 6 + c = 0 Eample 8: 4 + c = 0 Eample 9: c = 0 Eample 10: c = 0 Find two possible missing terms so that the epression is a perfect square trinomial. Eample 11: Eample 1: Eample 13: Eample 14: Reflection: 10

11 Solve a Quadratic Equation by Completing the Square. 1. Divide each side by a. Put constant on right side of equation. 3. Add b to both sides. 4. Write left side as a binomial squared. 5. Square root each side. (±) 6. Solve for the variable. E: Solve by completing the square. Step One: Divide by a Step Two: Put constant on right side of equation Step Three: Complete the square (add b to both sides) Step Four: Write left side as a binomial squared Step Five: Take the square root of both sides Step Si: Solve for the variable The solution set is 3 11, 3 11 Reflection: 11

12 Solve the equation by completing the square. Eample 15: = 0 Step One: Divide by a Step Two: Put constant on right side of equation. Step Three: Complete the square (add b to both sides). Step Four: Write left side as a binomial squared. Step Five: Take the square root of both sides. Step Si: Solve for the variable. or or Eample 16: = 0 Step One: Divide by a Step Two: Put constant on right side of equation. Step Three: Complete the square (add b to both sides). Step Four: Write left side as a binomial squared. Step Five: Take the square root of both sides. Step Si: Solve for the variable. or or Reflection: 1

13 Eample 17: + 66 = -18 Eample 18: = 0 Tell the step in which the mistake first occurs, fi the mistake, and solve the problem correctly. Eample 19: Milton showed the following steps in solving a quadratic equation by completing the square. In which step did Milton s first mistake appear? Fi the mistake and solve the problem correctly = 0 Step 1: + 14 = -4 Step : = Step 3: ( + 7) = 5 Step 4: + 7 = ± 5 Step 5: = -7 ± 5 Step 6: = ± Reflection: 13

14 Sample CCSD Common Eam Practice Question(s): 1. What are the solutions of A. = or = 4 B. = 10 or = 4 C. = 3 + i or = 3 i D. = 3 + i or = 3 i ?. Which is one of the appropriate steps in finding solutions for the square? A when completing B. C. D Sarah showed the following steps in solving a quadratic equation by completing the square. In which step did Sarah s first mistake appear? = 0 Step 1: - 10 = -11 Step : = Step 3: ( - 10) = 89 Step 4: - 10 = 89 Step 5: = A. Step Two B. Step Three C. Step Four D. Step Five Reflection: 14

15 Section 5.8 Quadratic Formula Objective(s): Solve quadratic equations by the quadratic formula. Analyze the nature of the roots of a quadratic equation. Essential Question: Describe the possible situations for the discriminant of a quadratic equation. What does a negative discriminant indicate and why? Homework: Assignment 5.8. #55 70 in the homework packet. Notes: Vocabulary The discriminant helps us find the number and type of solutions of a quadratic equation. The formula b b 4ac a is called the quadratic formula. Simplify Eample 1: Evaluate b 4ac for a =, b = 1, and c = 35 Eample : Evaluate b 4ac for a = 1, b = -4, and c = 4 Eample 3: Evaluate b 4ac for a = 1, b = 10, and c = 10 Eample 4: Evaluate b b 4ac for a = 1, b = -6, and c = 18 Reflection: 15

16 Eample 5: Simplify Eample 6: Simplify ( 3) 7 6 Eample 7: Simplify E: Solve the quadratic equation 8 1 using the quadratic formula. Step One: Rewrite in standard form (if necessary) Step Two: Identify a, b, and c. a 1, b 8, c 1 Step Three: Substitute the values into the quadratic formula. b b 4ac a Step Four: Simplify The solution set is 4 15,4 15 Use the quadratic formula to solve the equation. Eample 8: = 0 a = b = c = Reflection: 16

17 Eample 9: 6 = -9 a = b = c = Eample 10: + 38 = -14 a = b = c = Eample 11: = 0 a = b = c = Eample 1: = -1 a = b = c = Reflection: 17

18 Discriminant: The number under the square root in the quadratic formula. b 4ac The sign of the discriminant determines the number and type of solutions of a quadratic equation. If If If b 4ac 0, then the equation has two real solutions (two -intercepts). b 4ac 0, then the equation has one real solution (one -intercept). b 4ac 0, then the equation has two imaginary solutions (no -intercept). E: What is the discriminant of the quadratic equation type of solutions the quadratic equation has ? Give the number and Discriminant: b ac Since the disciminant is 0, there is one real solution. Consider only the discriminant, b - 4ac, to determine whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions eist. Eample 13: = 0 Eample 14: = 0 Eample 15: = 0 Sample CCSD Common Eam Practice Question(s): 1. How many real and imaginary solutions are there for the equation A. no real solutions, imaginary solutions B. 1 real solution, no imaginary solutions C. 1 real solution, 1 imaginary solution D. real solutions, no imaginary solutions ? Reflection: 18

19 . What is the solution set for the quadratic equation A. 3 3, ? B. 3 6, 3 6 C. 3 3,3 3 D. 3 6, What are the solutions of A. B. C. D. i i ? Reflection: 19

20 Section 5.9 Graphing Quadratic Functions Objective(s): Graph quadratic functions. Essential Question: Describe how h and k affect the graph of a quadratic function. Homework: Assignment 5.9. #71 78 in the homework packet. Notes: Vocabulary The graph of y = ( h) + k has verte (h, k). A quadratic equation is one that can be written in the form a + b + c = 0 where a, b, and c are real numbers and a is not 0. Fill in the table for the given function and plot the points on the graph. Eample 1: f() = y Eample : f() = y What did the 3 do to the graph? If you had f() = + 4, what would happen to the graph? Reflection: 0

21 Eample 3: f() = ( 1) y What did the 1 do to the graph? If you had f() = ( + 5), what would happen to the graph? End Behavior (Graphs) The appearance of a graph as it is followed farther and farther in either direction is called end behavior. For quadratics, the end behavior is indicated by drawing the positions of the arms of the graph, which may be pointed up or down. Quadratic End Behavior: 1. If the leading coefficient is positive, the right arm of the graph is up.. If the leading coefficient is negative, the right arm of the graph is down. When reading, we read left to right. When drawing a graph, we draw left to right. When PLANNING a graph, we must think RIGHT TO LEFT. The leading coefficient tells you whether the RIGHT side of the graph rises or falls. Sketch the graph of the quadratic function. Give the verte. Eample 4: + Eample 5: -( + 3) Verte: (, ) Verte: (, ) Reflection: 1

22 Eample 6: -( ) + 3 Eample 7: ( + 3) 7 Verte: (, ) Verte: (, ) Give the domain and range of the function. Assume the ends of the function continue on. Eample 8: Eample 9: Domain: _Range: _ Domain: _Range: _ Eample 10: Eample 11: Domain: _Range: _ Domain: _Range: _ Reflection: