Exponential Functions: (pp 296; 328) A function of the form y=a(b x ) were a 0 and 0<b<1 or b>1

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1 TABLE OF CONTENTS I. Day 1 Unit 5 Exponential Functions Orientation Day 2 Exponential Growth vs. Exponential Decay Day 3 Unit Five Lesson One Investigation One (5.1.1) Exponential Growth Day 4 Unit Five Lesson One Investigation One (5.1.1) Exponential Growth cont. Day 5 Day 6 Unit Five Lesson One Investigation Two (5.1.2) Compound Interest Day 7 Unit Five Lesson One Investigation Three (5.1.2) Compound Interest cont.. Day 8 Unit Five Lesson One Investigation Three (5.1.3) Compound Interest Day 9 Unit Five Lesson One Investigation Three (5.1.3) Compound Interest cont. Day 13 Unit Five Lesson Two Investigation One (5.2.1) More Bounce to the Ounce Day 14 Unit Five Lesson Two Investigation Two (5.2.2) Medicine and Mathematics Day 15 Unit Five Lesson Two Investigation Two (5.2.2) Medicine and Mathematics cont Day 16 Unit Five Lesson Two Investigation Four (5.2.4) Properties of Exponents II Day 17 Unit Five Lesson Two Simple Interest Exercise Day 18 Unit Five Lesson Two Compound Interest Exercise Day 19 Unit Five Assessment Review Day 20 Unit Five Assessment Review Day 21 Unit Five Exam Day 01 Unit Seven Quadratic Equations Day 02 Unit Seven Lesson One Investigation One (7.1.1) Pumpkin Chunkin cont. Day 03 Unit Seven Lesson One Investigation One (7.1.1) Homework 1,2,3,4 Day 04 Unit Seven Lesson One Investigation Two (7.1.2) Day 05 Unit Seven Lesson One Investigation Three (7.1.3) Day 06 Unit Seven Lesson One Investigation Three (7.1.3) cont Day 07 Unit Seven Lesson Two Investigation One (7.2.1) Day 09 Unit Seven Lesson Two Investigation Two (7.2.2) Day 10 Unit Seven Lesson Two Investigation Two (7.2.2) Cont. Day 11 Unit Seven Lesson Three Investigation One The Quadratic Formula.. Day 12 Unit Seven Lesson Three Investigation One The Quadratic Formula cont. Day 13 Unit Seven Lesson Three Investigation Two Examples Day The Quadratic Formula Derivation examples Day 15

2 Day 1 Unit 5 Exponential Functions Benchmarks: 2.1a Model nonlinear real world situations with equations/inequalities. 2.3a Solve problems using graphs, tables, and algebraic methods. Terms: Exponential Expressions: An algebraic expression in the form of b n, where b and n are real numbers or variables. The number b is called the base of the exponential expression, and n is called the exponent or the power. Exponential Functions: (pp 296; 328) A function of the form y=a(b x ) were a 0 and 0<b<1 or b>1 Example: Exponential Expressions 2 2 = 2*2 = b 2 = b*b = = 2*2*2 = b 3 = b*b*b = = 2*2*2*2 = b 4 = b*b*b*b = End of Day Day 2 Exponential Functions cont. Notice: The variable x is an exponent. As such, the graphs of these functions are not straight lines. In a straight line, the "rate of change" is the same across the graph. In these graphs, the "rate of change" increases or decreases across the graphs. Observe how the graphs of exponential functions change based upon the values of a and b: Example: When a > 0 and the b is between 0 and 1, the graph will be decreasing (decaying). For this example, each time x is increased by 1, y decreases to one half of its previous value. Example: When a > 0 and the b is greater than 1, the graph will be increasing (growing). For this example, each time x is increased by 1, y increases by a factor of 2. X Y= X Y=

3 10 10 Such a situation is called Exponential Decay. Such a situation is called Exponential Growth End of Day JAN05 Day 3 Unit Five Lesson One Investigation One (5.1.1) Exponential Growth Read Page 290 to understand the objective and desired outcomes of the Investigation. Essential Question- What are the basic patterns of exponential growth in variations of the Pay it Forward Process? 1. Follow Problem 1 on page 292 a. Graph b. c. 2. Follow Problem 2 on page 292 a. Tree Graph b. Table (emulate table in problem 1) Graph c. d. 3. Follow Problem 3 on page 292 a. NOW Next Rule Problem 1 NOW Next Rule Problem 2 b.

4 c. 4. Follow Problem 4 on page 292 a. b. c. d. TURN INTO INSTRUCTOR WHEN COMPLETE Day 6 Unit Five Lesson One Investigation Two (5.1.2) Getting Started Read Page 294 to understand the objective and desired outcomes of the Investigation. Essential Question- -What are the NOW-NEXT and y=? rules for basic exponential functions? -How can those rules be modified to model other similar patterns of change? 1. Follow Problem 1 on page 295 a. How many bacteria would exist after 8 hours (32 quarters) if the infection continues to spread as predicted b. 3 Number Qtr hrs Number Bacteria NEXT= Number Qtr hrs Number Bacteria NEXT= Number Qtr hrs Number

5 Bacteria NEXT= Number Qtr hrs Number Bacteria NEXT= Day 7 Unit Five Lesson One Investigation Two (5.1.2) Getting Started continued.. 4 Given: Now 4 people pay it forward 4a. Write the Next/Now Statement Next = 4b. Write the Rule N= 4c. How would the NOW-NEXT BE different if we started with 5 instead of four 4d. 5 Calculator problem: 5a. 5b. 6 X Y X Y X Y X Y 6a. 6b. 7: Given bacteria starts at 64 and doubles every hour 7a. What rule predicts the number at anytime x hours? 7b. What would it mean to calculate values for negative x? 7c. What would you expect for x= -1? -2?

6 -3? -4? 7d.explain 8 Explain problems a, c, e below End of Investigation Getting Started DAY 8 Investigation Compound Interest (Page 298) Use a calculator and record the following in your note book. What is going on: 1.1^2=? ^2=? ^2=? ^2=? What is going on? Essential Question- -How can you represent and reason about functions involved in investments paying compound interests? Given: You won the lottery and are given a choice to accept 10,000 dollars now or wait and collect in ten years. If you can invest the 10,000 in an 8% money market now, what will be the better of the two choices? 1.) Discuss the choices and write an explanation of your decision 2.) Write the rules a. For the next year and after (NOW-NEXT) 1. ANS. NEXT= NOW + NOW(.08) STARTING AT 10,000 OR NEXT = NOW(1+.08) STARTING AT 10,000---factoring out now OR NEXT = NOW(1.08) STARTING AT 10,000 b. After any year 1. ANS. Y = 10,000(1.08 X ) 3.) Use the rule from number 2b. to determine the value of the after 10 years 4.) Answer the following a. Describe the pattern ANS. The pattern is the growth starts at 800 per year and increases until nearly 1600 after ten years. b. Why isn t the change in CD the same every year?ans. The yearly interest is compounded, meaning last year s interest is included in the Next years calculation. c. How is the pattern of increase in CD balance shown in the shape of a graph or function? ANS. The graph gently curves upward and the curve slowly becomes steeper. d. 5.) Solve the following using the calculator table sub-routine

7 6.) Solve the following _ a. Given earning 4% annual interest compounded yearly b. Given 5000 earning 12% annual interest compounded yearly The following table illustrates each year for 10,000 at 8% Missed funeral 10, Initial Dollars 8% Year 8% percent Compounded 0 8% 8% 1 10, , , , , , , , , , , , , , , , , , , , , , , , , four days for a

8 25JAN11 DAY 13 INVESTIGATION More Bounce to the Ounce (page 323) Essential Question: What mathematical patterns in tables, graphs, and symbolic rules are typical of exponential decay relations? Do problems 1 only. For problem 2 explain only why the data will be the same or different then the calculated results? 1.) Fill in table if bounce is 2/3 of previous bounce. Bounce # Rebound Ht a.) How does the rebound height change from one bounce to the next? b.) Write the NOW/NEXT Statement c.) Write the Rule: y= d.) How will the data table, plot, and rules for calculating the height change if the ball drops first from 15 feet? 2. Explain why the theoretical data will be different or the same End of Investigation

9 DAY 14 Review Quiz Questions INVESTIGATION Medicine and Mathematics (page 326) Essential Question: How can you interpret and estimate or calculate values of expressions involving fractional or decimal exponents? How can you interpret and estimate or calculate the half-life of a substance that decays exponentially? 1. According to the graph on page 327 what appears to be the halflife?. 2. Answer question 2 on page 327 a. Graph the equation given. b. What do 10 and.95 represent? c. What percent is used up every minute? 3. Write the NOW/NEXT Rule 4. Answer problem 4 on page 327/328 a. 9.3 b. 8.0 c Answer problem 5 on page 328 a. Fill in the table Time (min) 19.5Units in blood b. Compare the data with your graph c. Estimate the following values and solve for x i. X is about 31.4 ii. X is about 4.3 iii. X < about Answer question 6 on page 328 a. At 15 units b. At 20 units c. At 25 units d. Explain the result

10 Day 15 Investigation Medicine and Mathematics cont Daily Starter Solve for Y(x) when x=4 Continue and finish Investigation End Investigation Begin Homework Page 338 #3,4 Problem 3 Page 338 Given the following: The steroid cyprionate is illegal in athletics and 90% will stay in your system one day after injection. 90% of the remaining will be in your system one day later and so on In this scinerio an athlete injects 100-mg of cyprionate. 1. Make a table showing the amount of drug in the system after every day for one week Time days Steroid % Plot Data and describe the pattern Describe pattern:

11 3. Write two rules (NOW/NEXT and y= ) for the problem. NEXT= y= 4. Use one rule to find steroids left after.5 days= 8.5 days= 5. Estimate the half-life 6. How long until steroid is less than 1% of its original strength? Penicillin The most famous antibiotic drug is penicillin. After its discovery in 1929, it became known as the first miracle drug because it was so effective on fighting serious bacterial infections. Drugs act somewhat differently on each person. But on average, a dose of penicillin will be broken down in the blood so that one hour after injection only 60% will remain active. Suppose a patient is given an injection of 300 milligrams of penicillin at noon. a. Write the rule in the form of y=a(b x ) that can be used to calculate the amount of penicillin remaining after any number of hours. b. Use your rule to graph the amount of penicillin in the blood from 0 to 10 hours. Explain what the pattern of that graph shows about the rate at which active penicillin decays in the blood. c. Use your rule to develop a table showing the amount of active penicillin that will remain at quarter hour intervals from noon to 5 p.m. i. Estimate the half life ii. Estimate the time it takes for an initial 300mg dose to decay so that only 10 mg remain active. d. If 60 % of a penicillin dose remains active one hour after an injection, what percent has been down in the blood.

12 31Jan2011 Day 16 Investigation Properties of Exponents Essential Question: What exponent properties provide shortcut rules for calculating powers of fractions, quotients of powers and negative exponents Find values of x and y that make the statement true. 1. Solve problem on page 332 a. b. c. d. 2. Examine problem 1 a. What pattern do you see? b. Don t answer c. What are some common problems when doing these problems? 3. Solve problem 3 on page 332 a. Z= b. Z= c. Z= d. x= e. x= y= f. Z= g. Z= h. Z= 4. Examine problem 3 a. What pattern do you see? b. Don t answer c. What are some common problems when doing these problems? Solve problem 6 on page 333: Given Insect growth is based on the formula: p=48(2^x). 7. Write an equivalent fraction that does not use exponents at all. a. b. c. d. e. f. g. h.

13 8. Examine the results of your work in Problems 6 and 7. a. How would you describe the rule defining negative integer exponents in your own words? b. What are some common errors in evaluating negative integer exponents? End of Investigation Check your Understanding Day 17 Simple Interest Exercise Simple Interest Equation A=total amount earned including principle P= initial amount invested r= rate of interest t = # of years Essential Question: What is Simple Interest? What is Compound Interest? What are the differences between the two? Simple Interest Whenever you borrow money, you pay a usage fee. That fee is called interest: Interest (I) = the amount charged for the use of borrowed money. The amount of interest you pay is based on three elements: the amount you borrow, the interest rate, and the length of time the money is borrowed for. The terminology for these elements is as follows: Principle (P): the amount borrowed Interest Rate (r): annual percentage of the principle that is charged as a fee Term (t): length of time the money is borrowed

14 When it is time to pay back the money, you are required to pay the principle plus the amount of interest that has accumulated. This is called simple interest and it is typically used for very shortterm borrowing or investments. The formula is as follows: Interest = Principle * rate * time (I=P*r*t) Example: If you borrow 1000 for five years at an interest rate of 10%, the amount of interest you pay is: I = P*r*t I= 1000*0.10*5 I= 500 The cost of borrowing 1000 for five years at 10% interest is 500. The total amount (A) due at the end of five years is principle + interest: A = P + I A = = 1500 An efficient way to calculate the total amount owed: A = P * (1+ rt) A = 1000 * ( *5) A = 1000 * 1.5 A = 1500 When you borrow money, you pay interest but when you invest money, you earn interest. An investment is really a case where you lend your money to someone else and they pay you interest. The same equations apply when calculating simple interest that is earned except now principle is the amount invested and interest is the amount earned. Example: If you put 5000 in a savings account that pays 2% annual interest, the amount of money you will have at the end of the year is: A = P * (1+ rt) A = 5000 * ( *1) A = 5000 * 1.02 A = 5100 The interest earned is 100. With simple interest you can set up an interest table that shows how much interest is accumulated over time. If we use the same example of 5000 in a savings account that earns 2% interest annually, the interest table looks like this: Term Interest Total (Yrs) After 5 years the savings account would have Suppose you want to figure out how long it would take to have 7250 in your savings account. You could create an interest table like the one above but a much better method is to use algebra and rearrange the simple interest equation to solve for time (t).

15 Two methods: 1. Calculate when the account equals 7250 A = P * (1+ rt) t=[1-(a/p)] / r t=[1-(7250/5000)] / 0.02 t=22.5 years 2. Calculate when 2250 interest has been earned (I = A-P = ) I = Prt t = I/(Pr) t = 2250/(5000*0.02) t = 2250/(100) t = 22.5 years Now suppose you want to know what interest rate would be needed to earn 7250 on a 5000 investment in 15 years. Two methods: 1. Calculate the rate that generates 7250 total A = P * (1+ rt) r=[1-(a/p)] / t r=[1-(7250/5000)] / 15 r= 0.45/15 r=0.03 = 3% 2. Calculate the rate that generates 2250 interest (I = A-P = ) I = Prt r = I/(Pt) r = 2250/(5000*15) r = 2250/75000 r = 0.03 = 3%

16 Student Worksheet The amount of interest you pay is based on three elements: the amount you borrow, the interest rate, and the length of time the money is borrowed for. 1. The amount charged for the use of borrowed money is called: a. Principle b. Term c. Interest d. Rate 2. The amount of an original investment is called: a. Principle b. Term c. Interest d. Rate 3. The formula for Simple Interest is: a. P=itr b. I=Prt c. r=i/pt d. Both b and c 4. The three elements used to calculate simple interest are,,. 5. How much interest does a 10,000 investment earn at 5.6% over 18 years? 6. Susan borrows 8650 to buy a used car and is charged 4.5% interest. If the term of her borrowing is 5 years, how much interest does she pay in total? 7. Henry invests 5000 in a mutual fund with an annual interest rate of 7.5%. He also has a 4-year, 10,000 loan at 3.75%. When will the amount of interest earned on the mutual fund be equal to the amount of interest paid on the loan? 8. If Sheila paid in interest on a 5 year loan of 5,800. What was the interest rate?

17 Day 18 Compound Interest Exercise Exponents, Compound Interest A special case of exponential functions is compounding or the increasing of money over time such that every time the invested money grows, the new amount is used to calculate interest. Compounding interest formula: A=total amount earned including principle P= initial amount invested r= rate of interest n= # of times compounded (per year) t = # of years 1. If Jack invests 5,000 in an account at 6% interest, a. Compounded annually, what is his investment worth in 5 years? b. Compounded monthly, what is his investment worth in 5 years? c. How many weeks are there in a year?? d. What is the answer if we compound the interest every week for five years? 2. Grandpa Henry isn't sure which investment is better. Should he invest in a fund which pays 3.7% compounded monthly or a fund which pays 6.0% compounded annually? What is the amount of both if 1000 is invested for a total of 10 years? 3. Aunt Hildegarde likes free gifts so when her bank gave away toasters, she invested 2,500 in an account that is compounded monthly at 3.2%. Unfortunately, Aunt Hildegarde was somewhat senile and she forgot about the account and when she died, you inherited it. If the money was untouched for 38 years, how much did you inherit?

18 14 Feb 2010 Day 01 Unit Seven Quadratic Equations APS Benchmarks 2.1a Model non-linear real-world situations with equations/inequalities 6.1a Use ratios, proportions, percents involving PSS Learning Objective: Define the Quadratic Equation as Lesson Pumpkins in Flight Lesson Objectives: What patterns of change appear in tables and graphs of (time, height) values for flying pumpkins and other projectiles? What Functions model those patterns of change? Problem 1: Fill in the table using the formula d=16t 2 Time(t) Distance Fallen (d) Height above Ground = = = Problem 2 Page 465 Use the date relating height and time to answer the following: a. What function rule shows the pumpkin height? y=100-16t 2 b. What equation can be solved to find the time when the pumpkin is 10 feet off the ground 90=16t 2 c. How long to drop to the ground100=16t 2 =2.5 d. How long to drop to the ground _h=75-16t t 2 = t 2 =0 Problem 3 Page 465 given pumpkin is shot straight up from a barrel at a point 20 feet above ground at a speed of 90 feet per second (about 60 mph) a. With no gravity how would the height change? The height would continue linearly up at a constant rate b. What function would relate height above the ground h in t seconds? h=20-90t c. How would you change the function in part b that shoots the pumpkin at 120 ft/sec. h=20+120t d. How would you change the function in part b that shoots the from 15 feet above ground? h=15+90t Problem 4 Page 466 a. Given: initial height = 20 ft, initial velocity is 90 ft/sec: Write the equation combining these conditions and the effect of gravity _h=20+90t-16t 2 b. How would you change the function for h= 15ft and initial velocity = 120 ft/sec? h=15+120t-16t 2

19 15 Feb 2010 Day 02 Unit Seven Quadratic Equations APS Benchmarks 2.1a Model non-linear real-world situations with equations/inequalities 6.1a Use ratios, proportions, percents involving PSS Learning Objective: Define the Quadratic Equation as Lesson Pumpkins in Flight continued. Lesson Objectives: What patterns of change appear in tables and graphs of (time, height) values for flying pumpkins and other projectiles? What Functions model those patterns of change? Problem 5 Page 466 a. b. Problem 6 Page 467 a. Problem 5 Page 466 Given h=h 0 +v 0-16t 2 a. What does the value of h 0 represent? what units are used? b. What does the value of v 0 represent? What units are used? Problem 6 Page 467 Given the function a. Suppose the initial height is 24 feet above ground at time = 0. What fact does that tell you about height as a function of time in flight? b. V 0 = Problem 7 Page 467 a. Plot the data graph and experiment with several values of v 0 and h 0

20 16 February 2011 Day 03 Unit Seven Quadratic Equations Homework Page 480 Problem #1,2,3,4 1. A first-time diver was a bit nervous about his first dive at a swimming pool. To ease his worries about hitting the water after a fall of 15 feet, he decided to push a tennis ball off the edge of the platform to see the, effect of landing in the water. a. What rule shows how the ball's height above the water h is related to elapsed time in the dive (t)? b. Estimate the time it will take the ball to hit the water. 2. Katie, a goalie for Riverside High School's soccer team, needs to get the ball downfield to her teammates on the offensive end of the field. She punts the ball from a point 2 feet above the ground with an initial upward velocity of 40 feet per second. a. Write a function rule that relates the ball's height above the field h to its time in the air (t). b. Use this function rule to estimate the time when the ball will hit the ground. c. Suppose Katie were to kick the ball right off the ground with the same initial upward velocity. Do you think the ball would be in the air the same amount of time, for more time, or for less time? 3. The opening of the cannon pictured at the left is 16 feet above the ground. The daredevil, who is shot out of the cannon, reaches a maximum height of 55 feet after about 1.56 seconds and hits a net that is 9.5 feet off the ground after 3.25 seconds. Use this information to answer the following questions. a. Write a rule that relates the daredevil's height above the ground h at a time t seconds after the cannon is fired. b. At what upward velocity is the daredevil shot from the cannon? c. If, for some unfortunate reason, the net slipped to the ground at the firing of the cannon, when would the daredevil hit the ground? 4. When a punkin' chunker launches a pumpkin, the goal is long distance, not height. Suppose the relationship between horizontal distance d (in feet) and time t (in seconds) is given by the function rule d = 70t, when the height is given by h = t - 16t 2. a. How long will the pumpkin be in the air? b. How far will the pumpkin travel from the chunker by the time it hits the ground? c. When will the pumpkin reach its maximum height, and what will that height be? d. How far from the chunker will the pumpkin be (horizontally) when it reaches its maximum height?

21 f 17 February 2011 Day 04 Unit Seven Quadratic Equations Investigation Page 469 Essential Question: How can tables, graphs, and rules for quadratic functions be used to answer questions about the situation they represent? What patterns of change appear in tables graphs of quadratic functions? Given: y=0.002x 2 -x Problem 1 use the function to answer the following a. What is the approximate height of the towers? b. What is the shortest distance from cable to surface? c. What interval is the height = to 75 feet? d. How is the graph similar? How is the rule similar? 2. Ticket example: a. What factors will affect the number? b. What kind of expenses will reduce the profit? 3. Given: Relationship of tickets sold (s) and price (x) is s=4,000-25x Expenses 1000 advertising, 3,000 rental, 1500 security, 2000 catering. a. Find the function for income (I) b. Find the function for profit (P) c. How do predicted values for I and P change with tickets ranging from 1 to 20? How are those patterns of change shown in graphs of profit and income? d. Which Ticket price gives the maximum income and profit? How many tickets will be sold at this price? e. What range of ticket prices should be considered to obtained the most people?

22 22 February 2011 Day 05 and Day 06 Unit Seven Quadratic Equations Investigation Page 473 Essential Question: How are the values of A, B, and C related to patterns in the graphs and tables of values for quadratic functions y=ax 2 +bx+c? 1. Do problem 1 on page 474 a. b. 2. Do problem 2 on page 474 a. What do they have in common? b. How are the patterns related? 3. Do problem 3 on page 474 a. Consider the function y=ax 2 when a>0 i. Why are the values of y always greater than or equal to 0? ii. Why are the graphs always symmetric curves with a minimum point (0,0)? b. Consider next the function y=ax 2 when a<0? i. Why are the values of y always less than or equal to 0? ii. Why are the graphs always symmetric curves with a maximum point (0,0)?

23 Day 07 Unit Seven Quadratic Equations Investigation Equivalent Quadratics Essential Questions: What strategies are useful in finding rules for quadratic functions? In deciding whether two quadratic expressions are equivalent? In deciding when one form of quadratic expression is more useful than another? Given: n=200-10x; Food = 5n; PROFIT = +INCOME FOOD COST DJ CLEANUP DJ = 150; PROFIT = Cleanup = 100 PROFIT = 1. Problem 1 page 493 a. Finish Table below b. Plot Income vs ticket cost Cost/ticket # of tickets sold Income Food Cost DJ Cost Security Clean-up Profit Problem 2 page 493 Discuss the question in class 3. Problem 3 page 493 Recall n=200-10x and c=5n therefore c=5(200-x) a. Therefore Profit = +x(200-10x)-5(200-10x) Profit = 200x 10x x Profit = -10x x 1250 (equation that is based solely on the cost per ticket. b. Discuss the question in class 4. Problem 4 page 494 Use calculator to practice graphing these equations. 5. Complete handout and turn in beginning of class tomorrow.

24 Day 09 Unit Seven Quadratic Equations Page 495 Investigation Reasoning to Equivalent Expressions Essential Questions: What strategies can we use to transform quadratic expressions into useful equivalent forms? 1. Solve the following problems on page 495 a. b. c. d. e. f. 2. Solve the following problems on page 496 a. b. c. d. 3. Solve the following problems on page 496 a. b. c. d. / End of Day / Day 10 Unit Seven Quadratic Equations Page 495 Investigation Reasoning to Equivalent Expressions Cont. Essential Questions: What strategies can we use to transform quadratic expressions into useful equivalent forms? 4. Solve the following problems on page 497 a. (x+5)(x+6) b. (x-3)(x+9) c. (x+10)(x-10) d. (x-5)(x+1) e. (x+a)(x+b) 5. Solve the following problems on page 497 a. b. c. d. e.

25 6. Solve the following problems on page 497 a. (x+4)(x-4) b. (x+5)(x-5) c. (3-x)(3+x) d. (x+a)(x-a) Do Homework problems Page 497 # 4,5,6,7,8 / END OF DAY

26 Day 11 Unit Seven Quadratic Equations Investigation The Quadratic Formula Essential Questions: What are some effective methods for solving quadratic equation algebraically? 1. Solve the following problems on page 512 a x = b. c. d. e. f. g. h.

27 2. Solve the Quadratic Equations algebraically on page 514 a.. b.. i. ii. iii. iv. v. vi. vii. i. ii. iii. iv.

28 Day 12 Unit Seven Quadratic Equations Investigation The Quadratic Formula Derivation Essential Questions: What are some effective methods and tools for solving quadratic equation algebraically? Given: 1. Isolate variables on one side 2. Divide both sides by a 3. Complete the square by taking half the linear coefficient, square it and add to both sides 4. Factor left side and simplify the right Take the square root of both sides And finally the quadratic formula is:

29 / End of Lesson / Day 13 Unit Seven Quadratic Equations Investigation The Quadratic Formula Derivation examples Essential Questions: N/A Solve for the following using any methods taught in this course then graph: 1. Roots = Y intercept = X(vertex)= Y(Vertex) = 2. Roots = Y intercept = X(vertex)= Y(Vertex) =

30 Day 14 Unit Seven Quadratic Equations The Quadratic Formula Derivation examples Essential Questions: N/A Solve the following questions and graph to scale labeling all determined points. Find Roots = Find Y intercept = Find X(vertex) X = Find Y(vertex) Y = Graph the function To scale

31 Practice Examples: Multiply through the problems

32 A quadratic equation is any equation that can be put in the standard form where a, b, and c are real numbers and The presence of the quadratic term ax 2 is what makes the equation quadratic, and therefore a cannot be zero. The presence of the linear term bx and the constant term c are not required. Therefore, b or c, or both, may be equal to zero. Not all quadratic equations appear initially in this standard form. Normally, though, you can identify a quadratic equation by inspection. Example: Which of the following equations are quadratic?

33 Roots of quadratic equations A solution, or root, of any equation is a value of the variable that makes the equation a true statement. The degree of the equation indicates the number of roots it may have. A quadratic, or second-degree, equation has two roots. These roots may be a pair of real numbers (in special cases, these numbers may be equal), or they may be a pair of imaginary numbers. Any root may be checked by substituting it into the equation to be certain that a true statement results. Example: For each equation, determine whether the given values are roots. 1. for x=3 and x=4 2. for x=2 and x=-2 Zero-Product rule Several methods can be used to solve quadratic equations. Those with rational roots can be solved by factoring. This method depends on one simple arithmetic fact called the zeroproduct rule. If the product of two or more factors is equal to zero, then at least one of those factors must be zero. NOTE: To use the zero-product rule, the quadratic equation must be in standard form, with zero on one side of the equation. Example:

34 Incomplete quadratics An incomplete quadratic equation is one that is missing either the linear or constant term. These can also be solved by factoring if the roots are rational. As the next examples show, only simple monomial factoring or difference of squares methods are needed. Example: