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Prerequisites What classes do you need to know: Algebra 1 Geometry Algebra 2 Precalculus is the combination of all previous mathematic classes PREREQUISITE #1 REAL NUMBER Review: Real Numbers • Real number: Any number that can be written as a decimal. • Activity: Match the corresponding vocab to its correct answer – Integers – Natural number(counting number) – Whole number {1,2,3…} {0,1,2,3…} {…-1,0,1…} Answer: – Integers – Natural number(counting number) – Whole number {1,2,3…} {0,1,2,3…} {…-1,0,1…} Review: Real Numbers • What are the difference rational numbers and irrational numbers? • Activity: Identify which of these are rational and irrational. – 1 , 𝑒1 , 11 3 5, -12, 1.75, 7.333… , 𝜋, 0, 8, 50 25 , log(6) Answer • Rational Number: number that either terminates or infinitely repeating – 1 , 11 3 -12, 1.75, 7.333… , 0, 8, 50 25 • Irrational Number: A number is infinitely nonrepeating – 𝜋, 𝑒 1 , 5, log(6) New!!! • {} this represent a set. It encloses the elements or objects. • Example: {0,1,2,3} • Translation: This set includes the solution of 0,1,2,3 Review: Real Number • Inequality ≥, ≤, <, > • Activity: Match the symbol with the answer • • • • a≥b a≤b a<b a>b a is less than b a is greater than or equal to b a is less than or equal to b a is greater than b Answer • • • • a≥b a≤b a<b a>b a is less than b a is greater than or equal to b a is less than or equal to b a is greater than b Bounded Interval Example Unbounded interval • Graph the following: – (3,∞) – (-∞, 3) – [3, ∞) – (-∞, 3] Properties of Algebra Commutative property Inverse property Addition: a+b=b+a Multiplication: ab=ba Addition: a+(-a)=0 𝟏 Multiplication: a*𝒂=1 , a≠ 𝟎 Associative property Distributive property Addition: (a+b)+c = a+(b+c) Multiplication: (ab)c=a(bc) a(b+c)=ab+ac (a+b)c = ac+bc Identity property Addition: a+0=a Multiplication: a*1=a a(b-c)=ab-ac (a-b)c = ac-bc Example: • Expand: (a+5)8 • Simplify: 9p+ap Answer • Expand: (a+5)8 – 8a+40 • Simplify: 9p+ap – p(9+a) Exponential Notation • 𝑎𝑛 = a*a*a*a*a… • a = base, n is the exponent • Example: – 57 = 5 ∗ 5 ∗ 5 ∗ 5 ∗ 5 ∗ 5 ∗ 5 – (−3)6 = −3 ∗ −3 ∗ −3 ∗ −3 ∗ −3 ∗ −3 – 15 − (𝑐𝑎𝑟𝑒𝑓𝑢𝑙 𝑤𝑖𝑡ℎ 𝑡ℎ𝑖𝑠 2 – Why is this the case? 𝑜𝑛𝑒)= −1 2 1 2 1 2 1 2 ∗ ∗ ∗ ∗ 1 2 Activity: Simplifying expressions • −3𝑎𝑏 7 𝑐𝑑 −2 8𝑎−6 𝑏2 𝑐 −1 𝑑𝑑 3 • 𝑥𝑦 2 𝑧 −3 𝑥 −2 𝑦 3 𝑧 9 • 𝑥𝑦 2 𝑧 −3 4 ( −2 3 9) 𝑥 𝑦 𝑧 Answer • −3𝑎𝑏7 𝑐𝑑 −2 8𝑎−6 𝑏2 𝑐 −1 𝑑𝑑 3 • 𝑥𝑦 2 𝑧 −3 𝑥 −2 𝑦 3 𝑧 9 • −24𝑏9 𝑑 2 ( ) 5 𝑎 𝑥3 𝑦𝑧 6 𝑥𝑦 2 𝑧 −3 4 ( −2 3 9 ) 𝑥 𝑦 𝑧 𝑥 12 𝑦 4 𝑧 24 Scientific notation • Scientific notation – related to chemistry where it is written as the product of two factors in the form a 10 n , where n is an integer and 1 a 10 • Activity: Convert the following into scientific notation or expand them out – – – – 3.54 x 109 1.29 x 10−7 0.000000459 4,970,000 Answer – 3.54 x 109 – 1.29 x 10−7 – 0.000000459 – 4,970,000 3,540,000,000 0.00000129 4.59 x 10−8 4.97 x 106 Homework Practice • Pgs 11-12 # 2-44e, 48-54e, 58-64e PREREQUISITE #2 CARTESIAN COORDINATE SYSTEM • Given the recent unemployment rate reports in California, describe the trend. What year represent the biggest increase? Decrease? What % can you predict about the future unemployment rate? (Graph it and answer) Year Unemployment rate 2006 4.9% 2007 5.4% 2008 7.2% 2009 11.3% 2010 12.4% 2011 11.8% 2012 10.5% Answer • Unemployment rate increasing from year 2006-2010 • Unemployment decreasing from year 20102012 • 2008-2009 post the biggest increase • 2011-2012 post the biggest decrease • 10%-12% unemployment rate • Scatter plot: plotting the (x,y) data pair on a cartesian plane • The previous question we just did is an example of a scatter plot What do you need to graph? • Coordinates – (X,Y), (input, output) – Example (3,-2) • Things to keep in mind: – Always label!!! Quick Talk: (30 second) • What is absolute value? Answer • Absolute Value: Always positive, tells distance. • Magnitude: size or distance Activity • l-4l = • l16-7l= • l9-27l = Answer • 4 • 9 • 18 Quick talk: How can you find the distance between two places/points? Answer: • Measure • Use distance formula Distance Formula • Distance Formula: It is derived from the Pythagorean theorem. D= 𝑥1 − 𝑥2 2 + 𝑦1 − 𝑦2 2 Activity: • Distance between (2,-5) and (-7,3) • Distance between (-1,-7) and (-6, 8) Midpoint Formula • Midpoint Formula: A formula to find the middle of the two points. 𝑥1 +𝑥2 𝑦1 +𝑦2 M=( , ) 2 2 Activity: • Find the midpoint between (6,8) and (-4,-10) • Find the midpoint between 1 3 5 7 (- , )𝑎𝑛𝑑 ( , ) 2 4 6 8 Review: Geometry Circle • Standard form of a circle: 𝑥−ℎ 2 + 𝑦−𝑘 2 = 𝑟2 – (h,k) is the center of the circle – r= the radius of the circle • Example: 𝑥 + 2 2 – (-2,9) is the center – 5 is the radius + 𝑦 − 9 = 25 Quick talk: How do you find the distance from the center to a point on a circle? Answer • By finding the radius • Use distance formula Homework Practice • Pgs 20-22 #5, 8, 9, 11, 13, 21, 23, 27, 31, 35, 39, 43, 49, 51, 55, 57 PREREQUISITE #3 SOLVING LINEAR EQUATIONS AND INEQUALITIES • How do you know if an equation/inequality is linear? Answer • If the highest power or the highest exponent is 1 Linear Equation • 𝑦 = 𝑚𝑥 + 𝑏 • Note: the highest power (exponent) is one Solving linear equation • Solve for x • 2 3𝑥 − 4 + 4 𝑥 + 1 = 8𝑥 + 3 Solve for y • 6𝑦−1 8 =3+ y 4 Solving inequality • Remember solving inequality is like solving an equation. • There are 3 ways of representing an answer – Example: • X>3 • 3, ∞ • graphically Solve • 2 𝑥 + 9 + 7 ≥ 5𝑥 + 16 Solve • 𝑧 4 + 1 2 < 𝑧 5 −3 Solving a sandwich • −7 < 2𝑥+15 3 <9 Note: Make 2 separate inequalities Homework Practice • Pgs 29-30 #3, 4-10, 17, 19, 25, 27, 39, 45, 51, 53, 55, 57 PREREQUISITE #4 LINES IN THE PLANE Quick Talk: How do you find slope? Answer • You find slope by taking the difference of the y divided by the difference of the x • Formula for finding slope: • 𝑦1 −𝑦2 𝑥1 −𝑥2 or ∆𝑦 ∆𝑥 • ∆= 𝑑𝑒𝑙𝑡𝑎, 𝑖𝑡 𝑚𝑒𝑎𝑛𝑠 "𝑐ℎ𝑎𝑛𝑔𝑒𝑠” or “differences” Activity • Find slope (-2,5) and (9,0) • (1,1) and (3,-4) Important!! • You can ALWAYS find an equation of a line when you are given 2 points. • What is the way of finding the equation of the line? Answer: Point-slope form • Point-slope form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) m=slope A point on the line (𝑥1 , 𝑦1 ) Example: 4 𝑦+8= − 𝑥−2 3 Slope = -4/3 Point on the line = (2,-8) Slope intercept form • Slope-intercept form 𝑦 = 𝑚𝑥 + 𝑏 𝑚 = 𝑠𝑙𝑜𝑝𝑒 𝑏 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 Example: Y=-3x+8 m=-3 b=8 y-intercept coordinate= (0,8) Using calculator • Graph the following: 1) 2x+3y=6 2) 𝑦 = −4 ( )𝑥 3 +9 Quick talk: The difference between parallel and perpendicular lines Answer • Parallel Lines – Slopes are the same, but different y-intercept – Example: – 𝑦 = 3𝑥 − 5 𝑦 = 3𝑥 + 98 • Perpendicular lines – Slopes are opposite reciprocal. – The intersect form 90 degree angle. – Example: – 𝑦 = 9𝑥 + 8 𝑦 = (−1/9) + 61 Solve it with your group raise your hand when your group has an answer • (3,8) and (4,2) • 1) Give me an equation parallel to those points • 2) Give me an equation perpendicular to those points Answer • 1) m=-6 • 2) m=1/6 Cracking word problems 1) 2) 3) 4) Know what you are solving for Gather facts about the problem Identify variables Solve Ultimate problem • In Mr. Liu’s dream, he purchased a 2014 Nissan GT-R Track Edition for $120,000. The car depreciates on average of $8,000 a year. 1) Write an equation to represent this situation 2) In how many years will the car be worth nothing. Answer 1) y=price of car, x=years y= −8000𝑥 + 120000 2) When the car is worth nothing y=0 X=15, so in 15 years, the car will be worth nothing. Ultimate problem do it in your group (based on 2011 study) • When you graduate from high school, the starting median pay is $33,176. If you pursue a professional degree (usually you have to be in school for 12 years after high school), your starting median pay is $86,580. • 1) Write an equation of a line relating median income to years in school. • 2) If you decide to pursue a bachelor’s degree (4 years after high school), what is your potential starting median income? Answer • 1) y=median income, x=years in school Equation: y= 4450.33x+33176 2) Since x=4, y=50,977.32 My potential median income is $50,977.32 after 4 years of school. Homework Practice • Pgs 40-42 #1, 5, 9, 13, 19, 23, 27, 35, 37, 43, 45, 53, 57, 59 PREREQUISITE #5 SOLVING EQUATIONS GRAPHICALLY, NUMERICALLY AND ALGEBRAICALLY Quick Talk:How do you know if you have a quadratic? Answer • When the highest power or highest exponent is 2 • Example: 𝑦 = 𝑥 2 − 6𝑥 + 9 What are the different ways of solving quadratic? • • • • 1) Factor (very important) 2)Quadratic formula 3) Completing the square 4) Graphing using calculator Solve using factor • 2𝑥 2 − 3𝑥 − 2 = 0 Solve using quadratic equation • Quadratic equation: 𝑥 = • Solve 2𝑥 2 − 3𝑥 − 2 = 0 −𝑏± 𝑏2 −4𝑎𝑐 2𝑎 Completing the square • Why is completing the square useful? It is useful because it is in vertex form. It makes it easy to graph. • Completing the square: • 𝑥 2 + 𝑏𝑥 + • 𝑥− 𝑏 2 2 𝑏 2 2 =𝑐+ =𝑐+ 𝑏 2 2 𝑏2 4 • Example: 4𝑥 2 + 20𝑥 + 17 = 0 Solve by graphing calculator • 4𝑥 2 + 20𝑥 + 17 = 0 Those were cake…now for the fun stuff • In your group, talk about which ways can you use to solve for the following and actually solve them. • 𝑥 3 − 2𝑥 − 1 = 0 • 𝑥 4 + 3𝑥 3 − 2𝑥 + 5 = 0 Answer • Graphing calculator • P/Q way (remainder theorem) How do you solve for absolute value? • l2x-7l=10 Homework practice • Pgs 50 #1-43odd PREREQUISITE #6 COMPLEX NUMBERS Quick Talk: • When you use the quadratic formula, you have a radical 𝑏 2 − 4𝑎𝑐. • What if 𝑏 2 − 4𝑎𝑐 > 0? • What if 𝑏 2 − 4𝑎𝑐 = 0? • What if 𝑏 2 − 4𝑎𝑐 < 0? Answer • 2 real solutions • 1 real solution • 2 imaginary solution or 0 real solution What is imaginary number? • Imaginary number is when you deal with 𝑖 • Very important to know: −1 = 𝑖 𝑖1 = 𝑖 𝑖 2 = −1 𝑖 3 = −𝑖 𝑖4 = 1 Remember, it is a cycle of 4. It just repeats itself. Example 1: Find 𝑖 58 Example 2: Find 𝑖 97 Example 3: Simplify −81 Example 4: Simplify −24 Answer 58 • = 14 𝑅 2 4 So that means it completed the cycle 14 times, with 2 left over. So the answer is 𝑖 2 = −1 97 • = 24 𝑅 1 4 So that means it completed the cycle 24 times, with 1 left over. So the answer is 𝑖 1 = 𝑖 • 9𝑖 • 2𝑖 6 Complex number: where real meets imaginary • 𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝑁𝑢𝑚𝑏𝑒𝑟 𝑓𝑜𝑟𝑚: 𝑎 + 𝑏𝑖 • 𝑎 = 𝑟𝑒𝑎𝑙 𝑝𝑎𝑟𝑡 𝑏 = 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 part Remember all algebraic operations are the same!!! It doesn’t change • Example: 7 − 4𝑖 + −3 + 17𝑖 = • Example: 2 − 𝑖 − 9 + 21𝑖 = Answer • 4 + 13𝑖 • −7 − 22𝑖 Multiply complex numbers • 7 − 4𝑖 ∗ −3 + 17𝑖 = • 2 − 𝑖 ∗ 9 + 21𝑖 = Answer • −21 + 119𝑖 + 12𝑖 + 68 = 47 + 131𝑖 • 18 + 42𝑖 − 9𝑖 + 21 = 39 + 33𝑖 Dividing complex number • Important to note: you can NEVER have the denominator in imaginary form or radical. • If the denominator is in imaginary form, you have to multiply the numerator and denominator by its conjugate. Example: if you have 𝑎 + 𝑏𝑖, the conjugate is 𝑎 − 𝑏𝑖 Finding conjugates • 5−𝑖 = • 3 + 2𝑖 = • 10 − 𝑖 = Answer • 5+𝑖 • 3 − 2𝑖 • 10 + 𝑖 Simplify • Simplify 1 5+2𝑖 Answer • Since the denominator is 5 + 2𝑖, the conjugate is 5 − 2𝑖. You have to multiply the top and bottom by 5 − 2𝑖. So the answer is 5−2𝑖 29 Simplify • 3 2−𝑖 • 5+𝑖 2−3𝑖 = = Answer • 6+3𝑖 5 • 7+17𝑖 13 Homework Practice • Pgs 57-58 #1-43odd PREREQUISITE #7 SOLVING INEQUALITIES ALGEBRAICALLY AND GRAPHICALLY Solve (show all types of answers) • 𝑥 − 8 > 18 Solve (show all types of answers) • 2𝑥 − 9 ≤ 15 Solve (show all types of answers) • 2𝑥 2 + 7𝑥 − 15 ≥ 0 Solve (show all types of answers) • 2𝑥 2 + 3𝑥 < 20 Solve (show all types of answers) • 𝑥 2 + 2𝑥 + 2 < 0 Solve (show all types of answers) • 𝑥 3 + 2𝑥 2 − 1 ≥ 0 Homework Practice • Pgs 64 #1-29 odd, 33, 37