1. Warm Up 1. Find the vertex, AOS, y-intercept and roots of y = -2x 2 – 8x – 10.

2. To find the solutions/roots/x-intercepts/zeros: ◦ Step 1: Set Equation = 0 ◦ Step 2: Let Y1 = 0 ◦ Step 3: Let Y2 be the quadratic portion of the equation. ◦ Step 4: Press Graph ◦ Step 5: Press Trace to one of the x – intercepts ◦ Step 6: Press 2 nd calc, 5, Enter, Enter, Enter. ◦ Step 7: Repeat 5 – 6 for other intercept Solving by graphing

3. x 2 – 2x = 15 x 2 + 12 = 0 Examples:

4. z 2 + 7z = -10 x 2 = 10x – 24 You try!

5. Quadratic Modeling

7. Parabolas in Real Life!! http://www.youtube.com/watch?v=cXOcB ADMp6o http://www.youtube.com/watch?v=cXOcB ADMp6o

10. 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

11. 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. a) What is the maximum height of the ball? How long does it take to reach the maximum height?

12. To find maximum height: Are we looking for x or for y? Graph the function. Adjust x min and x max, then press ZOOM 0. Find the vertex. h = -16t 2 +40 t +6

13. 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? What is her maximum height?

14. 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?

15. 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)

16. To find a time given height… Let y 2 = given height. ◦ y = 10 Find the intersection of y 1 and y 2 ◦ 2 nd, Trace, 5, Enter, Enter, Enter

17. 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?

18. 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?

19. 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?

20. To find the time it takes it hit the ground… This is asking us when does the height = 0? So what are we trying to do here? Let y 2 = 0. Find the intersection of y 1 and y 2

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?

24. To find the initial height… Find 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. Summarize