Warm Up: Choose Any Method We Have Discussed

Review!!! What is the quadratic formula? What form must the quadratic be in to use the formula? Must equal 0 Which part of the quadratic formula is the discriminant and what does it tell you? Positive – 2 solutions, 0 – 1 solution, Negative – no real solutions

Let’s Practice! Use the discriminant to determine the number of real solutions and then use the quadratic formula to find the solution(s): 1. y = x2 – 6x + 16 1. y = x2 – 6x + 6

Quadratic Word Problems October 9th

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

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

Example 1: Basketball 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 = -16t2 +40 t +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…

Basketball (continued) 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.

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.

Example #2: Underwater Diving 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) = -16t2 +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?

Example #3: Rocket The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t2 + 80t. a. What is the highest point that the rocket reaches? When does it reach this point? 32 ft and .8 seconds

Example #1: Basketball Let’s look again at our example #1 with basketball. (b) What if we wanted to find when the shot reached a height of the basket (10ft)? What did we say the max height of the ball was? How should we approach this problem?

To find a time at a given height… Set the equation equal to the height you want to be at Let y2 = given height Let y1 = the original equation Find the intersection of y1 and y2

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 Standard form: y = ax2 + bx + c Standard form: h = at2 + bt + c

Example #2: Underwater Diving Let’s look back to our diver. 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) = -16t2 +8t +30. c) When will the diver reach a height of 2 feet?

Example #3: Rocket The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t2 + 80t. b) At what time(s) is the rocket at a height or 25 m? .43 second and 1.17 seconds

Warm Up At a swim meet, the swimmer dives form the diving board. Her position above the water is represented by the equation h(t) = -16t2 + 24t + 40, where t represents time in seconds and h(t) represents the height in feet. What is the greatest height Janet reaches in her dive? How long does it take to reach the max height? When will Janet reach a height of 20 ft?

Example #1: Basketball Let’s look again at our basketball game: h = -16t2 +40 t +6. c) When will the ball hit the floor if it missed the basket entirely? How do we approach this problem…

To find the time it takes it hit the ground… This is asking us when does the height = 0 This time let y2 = 0. Find the intersection of y1 and y2

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

Example #2: Underwater Diving Let’s go back to our diver: h(t) = -16t2 +8t +30 d) When will the diver hit the water?

Example #3: Rocket The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t2 + 80t. c) When will the rocket hit the ground?

Example #1: Basketball Let’s go back to the game! h = -16t2 +40 t +6. d) What is the height of the ball when it leaves the player’s hands? How do we approach this problem..

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!

Example #2: Underwater Diving Let’s go to the water! h(t) = -16t2 +8t +30. e) How high is the diving board?

Example #3: Rocket The height, H meters, of a rocket t seconds after it is fired vertically upwards is given by h(t) = -50t2 + 80t. c) What was the initial height of the rocket?

Example #1: Basketball Game Time … One last time! h = -16t2 +40 t +6. e) What is the height of the ball after 2 seconds? How do we approach this problem..

Evaluating the Problem 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

Example #2: Underwater Diving Underwater one last time! h(t) = -16t2 +8t +30. f) How high is the diver after 1.5 seconds?

Summarize

Challenge Problem

Word Problem Posters You will be given a quadratic word problem In groups of three you will make posters for you problem Your poster should have: The word problem The solutions to each question A picture of the quadratic (graph) A picture that represents your story