Warm Up: Choose Any Method We Have

Discussed

Homework

Homework

Homework

Homework

Homework

Homework

Review!!!What is the quadratic formula?

What form must the quadratic be in to use the formula?

Which part of the quadratic formula is the discriminant and what does it tell you?

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 modelingWe 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 heightTime it takes to reach the groundInitial heightHow 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…

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.

Basketball (continued)

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.

Interpreting the Question …

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: RocketThe 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?

Example #1: BasketballLet’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 heightLet 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 + cStandard 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: RocketThe 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?

Example #1: BasketballLet’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: RocketThe 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: BasketballLet’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: RocketThe 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: BasketballGame 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 ProblemI take the x-value (time) and plug it

in to find the y-value (height)

h(2) = -16(2)2 + 40(2) + 6 = ____ feet

Underwater one last time!h(t) = -16t2 +8t +30.

f) How high is the diver after 1.5 seconds?

Example #2: Underwater Diving

Summarize

Challenge Problem

Word Problem PostersYou will be given a quadratic word

problemIn groups of three you will make posters

for you problemYour poster should have:1. The word problem2. The solutions to each question3. A picture of the quadratic (graph)4. A picture that represents your story