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Unit 9 – Quadratics 1

Name: ____________________ Teacher: _____________ Per: ___

Unit 1

Unit 2

Unit 3

Unit 4

Unit 5

Unit 6

Unit 7

Unit 8

Unit 9

Unit 10

– Unit 9a – [Quadratic Functions]


To be a Successful Algebra class,

TIGERs will show…

#TENACITY during our practice, have…

I attempt all practice I attempt all homework I never give up when I don’t understand

#INTEGRITY as we help others with their work, maintain a…

I always check my answers I correct my work, I never just copy answers I explain answers, I never just give them

#GO-FOR-IT attitude, continually…

I write down all notes, even if I’m confused I remain positive about my goals I treat each day as a chance to reset

#ENCOURAGE each other to succeed as a team, and always…

I offer help when I understand the material I push my teammates to reach their goals I never let my teammates give up

#REACH-OUT and ask for help when we need it!

I ask my questions during homework check I ask my teammates for help during practice I attend enrichment/tutorials when I need to

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Unit Calendar

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY

March 16 17 18 19 20

Domain and Range for Discrete and

Continuous Functions

Introduce Quadratic Graph and

Transformations

Identify Key Features from the Graph QUIZ

Identify Key Features from the Table

Identify Key Features from the Calculator

23 24 25 26 27

Mixed Practice Applications

Review

TEST A

DLA

30 31 March 1 2 3

English I EOC

Solve by Factoring Solve by Factoring Solve by Quadratic Formula

Holiday

6 7 8 9 10

Solve all 4 Ways

Solve Practice

QUIZ

Classify Functions Review

TEST B

Essential Questions

What are the similarities and differences between a linear and quadratic function?

What do zeroes, solutions, roots, and x-intercepts have in common? How do they differ?


Critical Vocabulary

Quadratic

Parabola

Roots

Zeroes

x-intercepts

Solutions

Vertex

Axis of Symmetry


Domain and Range: Discrete and Continuous

Discrete A relation/function/situation that is represented by _____________.

Continuous A relation/function/situation that is represented by a _____________.

Domain For a discrete relationship, it is a list of the ____ _________. For a continuous relationship, it is written as an inequality from Unit 2:

Range For a discrete relationship, it is a list of the ____ _________.


Reminders: Discrete

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

domain:

range:

Find the range of this function for the given domain.

15)( 2 xxf

domain: { -4, 1, 5 }

range:

domain:

range:

x y

-3 5 -2 5

-1 5

0 5

1 5

2 5

domain:

range:


Examples: Continuous

Graphs that continue onward in at least one direction:

It is possible for either the domain or the range to include every possible number, which we write as for “All Real Numbers”… otherwise, it is written as an inequality.

D: R:

D: R:

D: R:

Graphs that have the domain and/or range bounded on both sides: When bounded on both sides, we use a special double inequality to demonstrate the two endpoints that the x- or y-values fall between.

D: R:

D: R:

D: R:


D: R:

D: R:

D: R:

D: R:

D: R:

D: R:

D: R:

D: R:

D: R:

D: R:

D: R:

D: R:



Quadratic Function: The Parabola and Transformations

Parent Functions: The most BASIC form of a graph

Linear y = x

Quadratic y = x2

x f(x)

-3

-2

-1

0

1

2

3

x f(x)

-3

-2

-1

0

1

2

3

y = mx + b

Changes how ____ - intercept _______ or ________ the line is.

y = ax2 + c

Changes how ____ - intercept __________ or ______ the parabola is.


Examples: What happens when we change “c” in ax2 + c?

y = x

f(x) = x + 1

y = x + 3

f(x) = x – 5

y = x2

f(x) = x2 + 1

y = x2 + 3

f(x) = x2 – 5

We notice that the “c” is the ___ - _____________.

What happens when we change “a” in ax2 + c?

f(x) = - x

y = 2x

f(x) = 1

2 x

y = - 4x

f(x) = - x2

y = 2x2

f(x) = 1

2 x2

y = - 4x2

We notice that when “a” is _____________ the graph opens ____________.

We notice that when “a” is _____________ the graph opens ____________.

Ignoring the positive or negative sign

We notice that when “a” is ____________ than 1, the parabola is ____________.

We notice that when “a” is ____________ than 1, the parabola is ____________.


Practice:

Equation Width Translation Reflection

𝑦 =1

8𝑥2 + 5

Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change

𝑦 = −4𝑥2 Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change

𝑦 = −𝑥2 − 27 Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change

𝑦 = −3𝑥2 − 8 Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change

𝑦 =3

2𝑥2 − 6


𝑦 = 2𝑥2 + 4 Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change

𝑦 =1

4𝑥2


Put the 7 equations above in order from Narrowest to Widest: ___________, ___________, ___________, ___________, ___________, ___________, ___________,

If you translate the equation 𝑦 = 2𝑥2 + 4 up 2 units, what is the new equation? ______________. If you translate the equation 𝑦 = −𝑥2 − 27 down 3 units, what is the new equation? ______________. If you reflect the equation 𝑦 =

3

2𝑥2 − 6 , what is the new equation? ______________.

Write an equation that is wider than 𝑦 =

1

4𝑥2 ______________.



Quadratic Functions: Key Features from the Graph

Quadratic y = x2

y = ax2 + c

Changes how ____ - intercept __________ or ______ the parabola is.

Examples:

f(x) = -x2 + 4

Vertex: Maximum / Minimum? Axis of Sym: Roots: Domain: Range: Transformations:

y = 3x2 – 3

Vertex: Maximum / Minimum? Axis of Sym: Zeroes: Domain: Range: Transformations:

y = 4x2 + 1

Vertex: Maximum / Minimum? Axis of Sym: x-Int: Domain: Range: Transformations:

f(x) = −1

2x2

Vertex: Maximum / Minimum? Axis of Sym: Solutions: Domain: Range: Transformations:

Vertex: Maximum / Minimum? Axis of Sym: Roots: y-int: Domain: Range:


Practice:

Vertex: Maximum / Minimum? Axis of Sym: Roots: y-int: Domain: Range:

Vertex: Maximum / Minimum? Axis of Sym: Zeroes: y-int: Domain: Range:

y = -x2 + 1

Vertex: Maximum / Minimum? Axis of Sym: Roots: Domain: Range: Transformations:

f(x) = 4x2 – 4

Vertex: Maximum / Minimum? Axis of Sym: Zeroes: Domain: Range: Transformations:

y = -5x2 + 5

Vertex: Maximum / Minimum? Axis of Sym: x-int: Domain: Range: Transformations:

f(x) = x2 + 4

Vertex: Maximum / Minimum? Axis of Sym: Solutions: Domain: Range: Transformations:


Quadratic Functions: Key Features from the Table

x y -2 5

-1 0

0 -3

1 -4

2 -3

3 0

4 5

Vertex: Axis of Sym:

Roots:

0 X S

Examples:

f(x) = -x2 + 6x – 5

x f(x) 0 -5

1 0

2 3

3 4

4 3

5 0

6 -5


Roots:

y = x2 – 10x + 25

x y 2 9

3 4

4 1

5 0

6 1

7 4

8 9


x-intercept(s):

g(x) = x2 + 2x – 3

x g(x) -4 5

-3 0

-2 -3

-1 -4

0 -3

1 0

2 5

Vertex: Solutions when g(x) = 0: Axis of Sym: Solutions when g(x) = 5:

h(x) = x2 – 4x

x h(x) -2

-1

0

1

2

3

4

Vertex: Solutions when h(x) = 0: Axis of Sym: Solutions when h(x) = -3:


Practice:

f(x) = x2 – 8x + 7

x f(x) 1 0

2 -5

3 -8

4 -9

5 -8

6 -5

7 0


Roots: Solutions when f(x) = -8

y = -x2 – 6x – 9

x y -6 -9

-5 -4

-4 -1

-3 0

-2 -1

-1 -4

0 -9


Zeroes: Solutions when y = -4

g(x) = -x2 – 2x

x g(x) -5 -15

-4 -8

-3 -3

-2 0

-1 1

0 0

1 -3

Vertex: x-intercepts: Axis of Sym:

h(x) = x2 + 6x + 8

x h(x) -5

-4

-3

-2

-1

0

1

Vertex: Solutions when h(x) = 0 Axis of Sym:

FACTOR!!! (NO, you can’t forget this) 𝑛2 + 11𝑛 + 10

2x2 + 11x + 14 26 19 10x x

𝑥2 + 14𝑥 + 48

25 17 6x x 24 16 15x x


Quadratic Functions: Key Features from the Calculator

Quadratics on the Graphing Calculator:

Calculator On (You can press ON to return to the HOME SCREEN at any time)

Press NEW DOCUMENT, select NO TO SAVE CHANGES, select GRAPH

Type in equation f1(x) = __________ , ENTER

Press CTRL T to bring up table (* press CTRL T again to remove the table if needed)

Find the Vertex and ROXS (roots, zeroes, x-intercepts, solutions) in the table

VERTEX: find where table becomes symmetrical

ROXS: find where y=0 or changes sign

Examples:

f(x) = x2 + 8x + 15

x f(x)

Vertex:

Roots:

y = -x2 + 4

x y

Vertex:

Zeroes:

g(x) = x2 - 16

x g(x)

Vertex: x-intercepts:

h(x) = -x2 – 4x + 5

Vertex:

Maximum / Minimum?

Axis of Sym:

Solutions:

Domain:

Range:


Practice:

f(x) = x2 – 16x + 63

x f(x)

Vertex:

Roots:

y = -x2 + 9

x y

Vertex:

Zeroes:

g(x) = -x2 + 2x - 3

Vertex:

Maximum / Minimum?

Axis of Sym:

Solutions:

Domain:

Range:

h(x) = x2 + 14x + 48

Vertex:

Maximum / Minimum?

Axis of Sym:

Solutions:

Domain:

Range:

FACTOR!!! (NO, you can’t forget this)

𝑛2 + 4𝑛 − 12

𝑎2 − 13𝑎 − 30

7x2 - 5x – 2

2x2 – 6x + 4

25 7 6x x 3𝑥2 + 15𝑥 + 18


Quadratic Functions: Key Features Scavenger Hunt

Examples:

Use the table to answer the questions.

x y

-3 12

-2 5

-1 0

0 -3

1 -4

2 -3

3 0

Axis of Symmetry: Vertex: Zeroes:

Given the following function,

𝑓(𝑥) = −𝑥2 − 13𝑥 − 40 What are the roots?

Given the following function,

𝑦 = −3𝑥2 + 12𝑥 + 15 What is the vertex?

Which of the following is the vertex of the graph of the equation 𝑦 = −𝑥2 − 4𝑥?

A. (-3,3) C. (-2,4) B. (-4,0) D. (0,0)

What are the x-intercepts of the graph of the equation 𝑦 = 𝑥2 + 5𝑥 − 4 ?

A (−𝟓. 𝟕, 𝟎) , (. 𝟕𝟎𝟐, 𝟎) C (−. 𝟕𝟎𝟐, 𝟎), (𝟓. 𝟕, 𝟎)

B (−𝟓. 𝟕, 𝟎) , (−. 𝟕𝟎𝟐, 𝟎) D (−. 𝟕, 𝟎), (𝟓, 𝟎)

Graph the equation

𝑓(𝑥) = 𝑥2 + 4𝑥 − 12 Roots:

Axis of Symmetry:

Vertex:

Domain:

Range:



Quadratic Functions: Application Problems

What maximum height did the rocket reach? __________

How many seconds was the rocket in the air? __________

From what height was the rocket launched? __________

For what interval of time was the rocket above 35 meters?

Between _____ and _____

For how long was the rocket above 35 meters? __________

What is the y-intercept ( ______ , ______ )?

What does this represent? ____________

What is the x-intercept ( ______ , ______ )?


What is the vertex ( ______ , ______ )?


What time did the rocket reach its maximum height? _____

What is the Domain?

What is the Range?

Practice: The graph represents the relationship between the height of a ball and the distance it traveled after being thrown.

Are the following statements True or False?

____The ball reaches a maximum height of 14 feet.

____The ball reaches its maximum height after traveling 14 feet.

____The ball was thrown from a height of 6 feet.

____It took longer than 30 seconds for the ball to hit the ground

____The axis of symmetry for this graph is y = 14.


The graph below shows the height of a baseball from the time it is thrown from the top of a building to the time it hits the ground.

About what height is the building? How long did it take for the ball to hit the ground? Between what times is the baseball 80 meters or more above the ground? How much time passes while the baseball is 80 meters or more above the ground? What was the maximum height of the ball?

A farmer wants to create a rectangular fence. He has 120 feet of fencing and plans to use his barn as one of the sides of the rectangle. Here is a graph of the length of one side and the area.

What is the maximum area he can achieve? How long is the side when he achieves this area? If he makes the side length 20 feet, what would be his area? If he wanted an area of 800 square feet, what two side lengths could he have used? What is the Domain? What is the Range?

The graph shows h, the height in meters of a model rocket, versus t, the time in seconds after the rocket is launched.

At what time does the maximum height occur? What is the maximum height of the rocket? About how long did it take for the rocket to land?

What is the Domain? What is the Range?



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