Punkin Chunkin https://www.youtube.com/watch?v=dg4hboje3ve https://www.youtube.com/watch?v=jkndb6hzlbq 1
What if we... Recreated the event and shot our own pumpkin out of a cannon to study the flight of the pumpkin. How would the graph of this flight look if we compared the height of pumpkin above the ground to the time it spends in the air. 1. Which variable is the dependent variable in this situation? 2. What will happen to the height of the pumpkin as time passes? 3. Sketch a graph of how you think the height will change over time. 4. Does your sketch cross the x or y axis? If so, what do those points represent for our pumpkin? 2
Unit 7 Investigation 1 d = 16t 2 3
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GET CALCULATORS!! Warm up 3/7 Which of these movies is best? Explain your reasoning OBJECTIVELY (facts, not opinions) a b c Budget Gross RT (Critic) RT (Audience) $175 m $325 m 26% 63% $165 m $236 m 90% 88% $ 65 m $76 m 87% 75% Suicide Squad Dr. Strange Pete's Dragon 5
Flying Objects 2 h = 16t + v t + h 0 0 h = h = 0 v = 0 16t 2 = 6
A baseball player is in right field and is trying to throw a runner out at home plate. The equation for the flight of the 2 ball he throws is given by: h = 16t + 45t + 6 1) How tall is the player? 2) How hard did he throw the ball? 3) Estimate when will the ball hit the ground. 7
Solving With the Calculator Enter equation into Y= Calc, 2:Zero The screen says "left bound", chose an x value that is to the left of where you are looking (1 works), hit Enter The screen now says "right bound" chose a value to the right of the y intercept then Enter The screen now says "guess", hit Enter again The screen should now read X = (answer) y = 0 and the cursor should be on your intercept. 2 h = 16t + 35t + 2 8
Desmos.com 9
Suppose an NFL kicker hit the opening kickoff with an initial velocity of 52 feet/second. Create an equation that could model the situation. Ask Yourself: 1) What was the initial velocity? 2) What was the initial height? Don't forget gravity! Otherwise your kick will go to the moon! h = 10
Warm up 3/9 Which of these equations do you think would more accurately model an NFL Punter's kick? Why? 2 2 h = 16t + 35t or h = 16t + 35t + 2 11
#1 & 3 part C #2 & 4 part B 1. Y= To get the equation in 2. Window to set the dimensions, if you cant see the top, change the 'ymax' 3. Trace 4. x =? y =? 12
Redo those problems, This time with the EXACT Answer 13
Warm up 3/10 Thomas is playing catch with his son, when he throws the ball it can be modeled by a quadratic function. Which of the two makes more sense for modeling this action (think intial velocity and height) 2 2 h = 16t + 15t + 15 or h = 16t + 15t + 6 If Thomas' son drops the ball, use the equation you selected and find out how long the ball will be in the air. 14
Transformations 2 2 h = 16t + 15t + 15 h = 16t + 15t + 6 In the window of you calculator set the following parameters: xmin: 1 ymin: 2 xmax: 2 ymax: 20 xscl: 1 yscl: 1 Then enter both equations into your "y=" (on separate lines), What appears to be different between the two graphs? 15
50 Shown is the graph of y = 16t + 45t 2 copy it to your notebook On the same axis sketch the graph of y = 16t + 45t + 8 Which of the two graphs would better represent an NBA player throwing an inbounds pass? 2 4 Using the equation you chose, find out how long the basketball will be in the air 16
75 One more 2 Shown is the graph of y = 16t + 65t + 4 On the same axis sketch the graph of y = 16t + 65t 4 8 Which of the two graphs would better represent a little league player throwing a baseball? Using the equation you chose, find out how long the basketball will be in the air 17
200 Warm up 3/13 Shown is the graph of y = 16t + 100t + 50 2 On the same axis sketch the graph of y = 16t + 100t 5 10 Which of the two graphs would better represent a model rocket launched from Jimmy's driveway. Using the equation you chose, find out how long the rocket will be in the air 18
Standard/Factored Forms 2 y = ax + bx + c y = (x + a)(x + b) 19
Expanding a Quadratic This idea is rooted in the distributive property, we just have two terms to distribute now. ex. Write (x + 4)(x 2) in standard quadratic form. Multiply inside, add/subtract after 20
Expanding a Quadratic 2 ex. Write (x 3) in standard quadratic form. 21
Homework 1. (x + 2)(x + 5) 2. (x + 3)(x 7) 3. (x 4)(x 6) 4. (x + 4)(x 4) 5. (x + 7)(x 9) 6. (x 5) ` 7. (x + 10) 8. 4(x + 1) 2x(3x + 4) 22
Warm up 3/16 Which equation would more accurately represent the flight of a homerun hit? Explain in at least two complete sentences. 2 2 h = 16t + 100t + 100 or h = 16t + 100t + 4 Then calculate when the ball will hit the ground. 23
Homework 1. (x + 2)(x + 5) 2. (x + 3)(x 7) 3. (x 4)(x 6) 4. (x + 4)(x 4) 5. (x + 7)(x 9) 6. (x 5) ` 7. (x + 10) 8. 4(x + 1) 2x(3x + 4) 24
Review 1) Explain which equation would better represent a submarine's missile launch. 2 2 h = 16t + 1000t 100 or h = 16t + 1000t + 40 Then calculate when the missile will hit the ground. 2) Expand the following a) x(x 9) b) (x + 4)(x 3) c) (x 8) d) x(20 x) + 3x(x 1) e) (2x 3)(3x + 1) 2 25
Warm up 3/20 Expand a) x(3x + 4) b) (2x + 4)(x 9) 26
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The Quadratic Formula b + b 2 4ac 2a 2 for y = ax + bx + c The quadratic formula allows us to find solutions (or zeros) of functions when we may not be able to use a calculator. 31
Example Solve: 2 y = x + 7x + 12 32
2 Solve: y = x 2x 8 Example 2 33
Solve: y = 2x 2+ 4x 1 Warm up 3/21 34
2 Solve: 0 = x + 4x Missing a Value? 35
Not Equal to 0? Solve: 9 = x 2 36
Assignment Due Thursday 2 2 1) 0 = x + 3x 10 2) 0 = 2x 2 2 2 3) 0 = x 16 4) 3 = x + 6x + 12 2 5) 0 = x + 1 37
Practice Two brothers Gary and Gerry are launching a model rocket. The equation relating the height of the rocket to the time after launch is given by: h = 2 + 125t 16t 2 1) How tall is the launch platform? 2) How fast is the rocket traveling when it leaves the platform? 3) Use your calculator to find the time when the rocket will touch down. 4) Estimate the time when the rocket will reach it's highest point. How high do you think it will get? 0 < x < 10 0 < y < 250 38
Calculating The Vertex Same process for calculating x intercepts (zeros) only instead of selecting "zero" in the calc menu, you will select "minimum" (if the parapbola opens upward) OR "maximum" (if the parabola opens downward) How far up did the rocket get? 39
Class Assignment Find the three values listed below for both situations Grounded Rocket Football Punt 40
Hw check Warm up 1/19 Need Calculator Aaron Rodgers has now managed to throw two insane "hail mary's" this year. The throw on Saturday to send the game into 2 overtime could be modeled by the equation h = 6.5 + 63t 16t 1) When did the ball reach it's maximum height? 2) What was the maximum height of the throw? 3) When would the ball have hit the ground if Jeff Janis hadn't caught it? 41
Calculating Height = 0 h = 6.5 + 63t 16t 2 When was the ball 40 feet above the ground? We've only looked at finding when an object hits the ground again or when height is equal to 0. In other terms 0 = 6.5 + 63t 16t 2 But how do we do: When "40" isn't in the calc menu 40 = 6.5 + 63t 16t 2 THERE WILL BE TWO ANSWERS 42
d. When was Jason 100 feet above the water? 43
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Warm up 1/20 Need Calculator Hw Check ~11:08 Jesse kicks a soccer ball off a penalty kick, the height of the ball over time is given by: h = 75t 16t 2 1) How long will the ball be in the air? 2) At what time(s) was the ball 20 feet off the ground? 45
Homework Which equation makes more sense to relate the height of the ball off of a home run over time in a baseball game? Why? A) B) h = 65t 16t 2 h = 4 + 65t 16t 2 46
FOIL / Box Method 3x (x + 4) 2 3x + 12x (x 4)(x + 3)? 47
FOIL Method (x 4)(x + 3) First Outside Inside Last 48
Box (Lattice) Method 49
Warm up 1/22 Expand (FOIL) 1) 2) (x 5) (x + 7) (2x + 4)(4x + 6) 50
page 497 #5 & 6 Practice 51
1:12 Warm up 1/25 HW Check 1) (x + 7)(x 7) 2) (x + 5) 2 52
Homework Find the income made from the following situations 1) Mr. G sells 18 notebooks for $0.25 each. 2) STUCO sells 450 tickets to "Bring on the Neon" for $8 each. 3) The cafeteria sells 80 bags of chips at $0.75 each. 4) What general rule would you make to calculate income for any sales situation? Income = 53
3. Miss Potter enjoy's baking and selling cakes. After several experiences, she has learned that is she sells her cakes for different prices per piece, p, she will have s number of customers where s=650 50p. Miss Potter's expenses include flour, butter, eggs, powdered sugar and vanilla. For each cake her expenses total $150. a. What is the equation that represents Miss Potter's income from her cakes? b. What is the equation that represents Miss Potter's profit from her cakes? c. What is the maximum price that Miss Potter should charge per piece to optimize her profit? (get the most profit) d. What is Miss Potter's break even point for selling her cakes? (What price(s) does she make 0 profit?) e. For what prices will she make $500? 54
Warm up 1/26 1) Expand: y = ( x + 3)(x 6) 2) What is the maximum value of the resulting function? 3) What are the zeros of the resulting function? 55
Warm up 2/2 Simplify or solve 1) 25 2 2) x = 100 2 3) x 81 = 0 4) x = 7 + 3 5) x = 3 + 5 2 56
Expressions + 57
Examples x= ( 2 + 4) 2 2 x = 15 + 2(7 2) vs x = 100 2 58
1)X= 5±8 Practice 2) X= 3±7 59
12 Warm up 2/3 Solve 1) 3 + ( ) 2 4 2) x = 3(30 3) = x 2 60
1)X= 5±8 Practice 2) X= 3±7 61
Warm up 2/8 1) 2) x = 62
The Quadratic Formula Ever wish you could find the zeros of a quadratic function without a calculator? Well now you can! Find out what prices to charge for a product or service! Calculate distances and flight patterns! Prepare yourself for success in CMIC 2! Our Function Must Be in the Form: 2 0 = ax + bx + c Where a, b and c are constants (numbers). Keep in mind that if a, b or c has a value of zero the function will look slightly different. Essentially we're saying "what x value makes my function equal to 0?" 63
For: Quadratic Formula 2 0 = ax + bx + c x = b + b [4(a)(c)] 2 2a 64
x = Example 2 0 = x + 4x 5 2 ( ) + ( ) [4( )( )] 2( ) a = b = c = 65
Pg 515 #2 66
Warm up 2/10 Solve: 6 = 5x + x 2 67
Finish p 515 #2 On your Handout 68
Warm up 2/16 2 Solve: 8 = x + 2x 69
Practice Sheet Answers 70
The Discriminant The discriminant is a part of the quadratic formula that lets us know how many solutions a particular quadratic has. 2 b [4(a)(c)] 71
Two Solutions The most common result for quadratics, this happens when the discriminant has a positive value. or 72
One Solution A quadratic has one solution when the discriminant has a value of zero or 73
No Solutions A quadratic has zero solutions when the discriminant is negative or 74
Homework How many solutions do each of the following expressions have? 2 1) 0 = x + 4x 5 2 2) 0 = x + 4x + 5 2 3) 0 = x + 6x + 9 2 4) 0 = x 25 2 5) 10 = x + 9x + 30 6) 5x = 14 + x 2 75
Warm up 2/17 2 Solve: x 16 = 0 76