1. Warm Up What are the three types of graphs you will see with quadratic linear systems? Sketch them & label how many solutions. Find the solution(s) to Given the points (-2, -27) (-1, -13) (0,-3) (1, 3 ) (4, -3) Find the quadratic equation!

2. Quadratic Word Problems! February 24 th, 2014

3. These are the types of questions you need to be able to interpret & answer!

4. Basketball parabola! http://www.youtube.com/watch?v=dSR WY5vUHCUhttp://www.youtube.com/watch?v=dSR WY5vUHCU Until 1:25

7. Quadratic modeling We can create quadratic functions to model real world situations all around us. We can use these models to find out more information, such as: Minimum/maximum height Time it takes to reach the ground Initial height How long it takes to reach a height

8. Example #1: For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40t +6. a)What is the maximum height of the ball? How long does it take to reach the maximum height? How do we approach this problem…

9. To find maximum height: Are we looking for x or for y? Graph the function. Adjust the window as needed. (this takes some practice!) Find the vertex.

10. Interpreting the question… The maximum or minimum HEIGHT is represented by the Y VALUE of the vertex. How long it takes to reach the max/min height is represented by the X VALUE of the vertex.

11. Example #2: The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. a)How long does it take the diver to reach her maximum height after diving off the platform? b)What is her maximum height?

12. Example #3: The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t 2 + 80t. a) What is the highest point that the rocket reaches? When does it reach this point?

13. Example #1: For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. b) When will the shot reach the height of the basket? (10 feet) How do we approach this problem…

14. To find a time at a given height… Set the equation equal to the height you want to be at Let y 2 = given height Let y 1 = the original equation Find the intersection of y 1 and y 2

15. Interpreting the problem… The X VALUE always represents TIME How long it takes…. So when you find the intersection, it should have X = time, and Y = height

16. Example #2: The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. b) When will the diver reach a height of 2 feet?

17. Example #3: The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t 2 + 80t. c) At what time(s) is the rocket at a height or 25 m?

18. Example #1: For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. c) When will the ball hit the floor if it missed the basket entirely? How do we approach this problem…

19. To find the time it takes it hit the ground… This is asking us when does the height = 0 Let y 2 = 0. Find the intersection of y 1 and y 2

20. Interpreting the problem… When asking when something HITS the GROUND you should think ZERO! GROUND = ZERO Find the second zero (not the first!) think left to right…goes up then down

21. Example #2: The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. c) When will the diver hit the water?

22. Example #3: The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t 2 + 80t. c) When will the rocket hit the ground?

23. Example #1: For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. d) What is the height of the ball when it leaves the player’s hands? How do we approach this problem..

24. Interpreting the problem…. Here we want to find the INITIAL HEIGHT….where did the ball start? ON the ground? In someone's hands? The INITIAL HEIGHT is the Y-INTERCEPT!

25. Example #2: The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. d) How high is the diving board?

26. Example #3: The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t 2 + 80t. c) What was the initial height of the rocket?

27. Example #1: For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t 2 +40 t +6. d) What is the height of the ball after 2 seconds? How do we approach this problem..

28. Evaluating I take the x-value (time) and plug it in to find the y-value (height) h(2) = -16(2) 2 + 40(2) + 6 = ____ feet

29. Example #2: The distance of a diver above the water h(t) (in feet) t seconds after diving off a platform is modeled by the equation h(t) = -16t 2 +8t +30. How high is the diver after 1.5 seconds?

30. Summarize

31. Challenge Problem – can you interpret what this problem is asking?

32. Challenge Problem –hint: use quadratic/linear systems!

33. Homework: Worksheet Complete the entire worksheet! You have 30 minutes in class Work on quiz corrections for ½ credit back