QUADRATIC

EQUATIONS

02 CBxAx

DO NOW

A salesperson receives a base salary of $35,000 per year and a commission of 10% of the total sales for the year.a) Write a model to represent this

situation. Define your variables.b) Graph your model.c) If the salesperson sells $250,000

worth of merchandise throughout the year, what will his/her total income be for the year.

d) How much merchandise would the salesperson have to sell in a year to earn $100,000 total income?

e) Justify your answers to parts c & d graphically.

THROWING A BASEBALL Lolita loves baseball.

She is playing catch with her brother.

What type of path will the ball follow? Draw a sketch.

Why is the path NOT linear?

Can you determine how high will the ball go? What does its maximum height depend on?

- How hard she throws it- The angle at which she throws it

Can you determine how long will the ball be in the air before her brother catches it?What if he doesn’t catch it, what will happen? UNDE

RSTA

NDIN

G TH

E SI

TUAT

ION

THROWING A BASEBALL Lolita is 4’ tall and throws the ball with

an initial velocity of 32in/sec at an 86angle.

How high do you think the ball will go?

How long will it take the ball to hit the ground?

PREDICTION

THROWING A BASEBALL This situation can be modeled by

the following quadratic equation:

Where h(t) is the height of the ball in inches at any time t seconds

after she releases the ball.

SKET

CH A

GRA

PH 483216)( 2 ttth

Let’s draw a sketch of the graph.

What should it look like?

THROWING A BASEBALL

What is the initial height of the ball when Lolita throws it?

Does this make sense in the equation?

Let’s find some values…How high will the ball be after ½ second?

1 second?1½ seconds?2 seconds?

2½ seconds?3 seconds?

3½ seconds?

Graph those points…Does the graph make sense in our situation?

CREA

TE A

TABL

E483216)( 2 ttth

00.51

1.52

2.53

3.5

THROWING A BASEBALL

What information can we gather from the table?What does the point represent?What does a height of 0 mean?

Do you see any pattern in the values?What do you think this means?

SYMMETRY!

What do you think is the maximum height the ball will reach?

Justify your reasoning.

ANAL

YZE

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0 480.51

1.52

2.53

THROWING A BASEBALL

Although cannot be negative in our situation, from the pattern we see, what other value would give a height of 0 ft?

Justify your reasoning. Does this make sense according to our graph?

How could you use the equation to mathematically find the value of the ball

when it has a height of 0ft?(the time it takes for the ball to hit the

ground) CALC

ULAT

E483216)( 2 ttth

THROWING A BASEBALL

Thinking about the symmetry seen in the table, how could we use the two “zeros” you just found to find the value at which

the ball is at its maximum height?

What does this value mean in context?

How could we use this value to find the maximum height the ball will reach?

How can you find , the height, for ?Justify your reasoning. IN

TERP

RET

483216)( 2 ttth

THROWING A BASEBALL

What is the initial height of the ball when it was thrown?

Mathematical process: ______________________

What is the maximum height the ball will reach?

Mathematical process: ______________________

How long will it take the ball to reach its maximum height?

Mathematical process: ______________________

When will the ball hit the ground?Mathematical process: ______________________

SUM

MAR

IZE

483216)( 2 ttth

DO NOW1) FACTOR fully:

2) A ball is thrown up from 3 m above the ground with an initial velocity of 14 m/s. The formula: can be used to model this situation, where is the height in meters, is the initial velocity, and is the initial position.a. Write an equation for the height of the ball.b. Find when the ball will hit the ground.c. Find how long it will take the ball to reach its maximum height and what that maximum height of the ball is.

DO NOW2) Imagine the path of a toy rocket after its launched. It can be modeled byt he equation where is the height of the rocket in inches and is time in seconds.

a. Graph your model (use Identify the domain & range.b. What is the initial height of the rocket? c. How high is the rocket after 2 seconds?d. When will the rocket hit the ground?

1) Solve:

EXTENSIONImagine the equation represents the path of a toy rocket after its launched, where is the height of the rocket in inches and is time in seconds. - Find when the rocket hits the ground mathematically.- Find how long the rocket will take to reach its maximum height and what that maximum height is mathematically.

DO NOW1) A diver is standing on a platform 24 ft. above the pool. She jumps from the platform with an initial velocity of 8 ft/s. Use the formula where is the height above the water, is the initial velocity, and is the initial position. - How long will it take for her to hit the water?2) You are trying to dunk a basketball. You need to jump 2.5 ft. in the air to dunk the ball. The height that your feet are above the ground is given by the function where is time in seconds. - What is the maximum height your feet will reach? - Will you be able to dunk the basketball?

VERTEX The maximum or minimum point on a parabola is

called its VERTEX.

* Note: a quadratic equation has a maximum/minimum value, but the vertex is always a POINT (x,y)!

Every quadratic equation has a vertex !

The vertex is always halfway between the parabola’s two ‘zeros’ because of symmetry.

AXIS OF SYMMETRY The axis of symmetry passes

through the vertex and folds the graph exactly in half. This imaginary line is called its AXIS OF SYMMETRY.

Every quadratic equation has an axis of symmetry!

Any two points equidistant from the axis of symmetry by a horizontal line will have the same y-value. These two points are called REFLECTED POINTS.

WHAT QUADRATIC EQUATION IS THIS?

The following represents the table of a quadratic equation.

- Identify the axis of symmetry. Justify your reasoning.

- Fill in the missing values.- Identify three (3) sets of

reflected points. - Use any 3 points to write a system of equations for this

scenario. - Solve the system to write an

equation for this model.- Use the equation to find the

vertex point.

x y = ???4 33

-12 673673

9 253253

-2 33

* Hint : what is the relationship between a point and the graph’s

equation?

𝒚=𝒂𝒙𝟐+𝒃𝒙+𝒄

WRITING THE EQUATION

𝑝𝑜𝑖𝑛𝑡 (4 ,33 )

𝑝𝑜𝑖𝑛𝑡 (−7 ,253 )

System of equations:

{ 16𝑎+4𝑏+𝑐=334 𝑎−2𝑏+𝑐=3349𝑎−7𝑏+𝑐=253

𝑦=𝑎𝑥2+𝑏𝑥+𝑐

𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑𝑝𝑜𝑖𝑛𝑡 (−2 ,33 )

x y = ???4 33

-12 673673

9 253253

-2 33

{ 12𝑎+6 𝑏=0−45𝑎+5𝑏=−220  

𝑎=4𝑏=−8

𝑐=1

Write an equation for the graph below using matrices.

* Hint: Find a 3rd point = a reflection of the given y-intercept.

WRITING THE EQUATION

WRITING THE EQUATION

𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (0 ,−5 )𝑣𝑒𝑟𝑡𝑒𝑥 (2 ,−9 )

Matrices:

𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑𝑝𝑜𝑖𝑛𝑡 (4 ,−5 )

[ 421−90 01−516 41−5]{ 4𝑎+2𝑏+𝑐=−9

0 𝑎+0𝑏+𝑐=−516𝑎+4𝑏+𝑐=−5

𝑦=𝑎𝑥2+𝑏𝑥+𝑐

[ 10 01010−4001−5 ]

YOU TRY IT

𝑣𝑒𝑟𝑡𝑒𝑥 (− 18 ,− 3618 )

A quadratic graph has two zeros at and .

The minimum value is Write an equation for this function.

[1−864−2888813616 025−10 40 ]

{164

𝑎− 18𝑏+𝑐=− 361

88116

𝑎+ 94𝑏+𝑐=0

254𝑎−5

2𝑏+𝑐=0

[ 1008010 2001−45]

* Hint: Clear Fractions!

HOMEWORK (HONORS)NUMBER THEORY – There are 2 numbers. Twice

the smaller number is 23 more than the other number. Twenty-five times the larger number is 62 less than the product of the two numbers. What are these two numbers?

EQUATIONS – Use the two points given on the graph below to write an equation for the given quadratic function.

pg. 234 #39, #51, #53 #39 also find max height #53: roots = zeros

HOMEWORKWrite a model for a quadratic graph that has zeros at and and whose maximum value is 8. Graph that model. How would the equation change if the minimum value were 8?Write a model for the graph below. Write your final answer in standard form:

(𝟎 ,𝟖)(𝟑 ,𝟒)

𝒚=𝒂𝒙𝟐+𝒃𝒙+𝒄