Academic CCM2 – Unit 4 – Quadratics Name: ______________________Notes and Activities Date: _____________ Pd: _____Unit Objectives:F-LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-SSE.1 Interpret expressions that represent a quantity in terms of its context.a. Interpret parts of an expression, such as terms, factors, and coefficients.b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Date Lesson ActivityThursday, October 30 Multiplying Polynomials

GCF and Factor by GroupingFriday, October 31 Differences of Squares and

Factoring TrinomialsMonday, November 3 Slide and Divide

Mixed Factoring ReviewTuesday, November 4 Solving Equations by FactoringWednesday, November 5 Unit 4 Quiz 1 Unit 4 Quiz 1Thursday, November 6 Graphs of QuadraticsFriday, November 7 Translating QuadraticsMonday, November 10 DiscriminantTuesday, November 11 NO SCHOOLWednesday, November 12 Quadratic FormulaThursday, November 13 Unit 4 ReviewFriday, November 14 Unit 4 Test Test

1

FRED Functions And Quadratics

POLYNOMIALS: FOIL BOX METHOD Part 1

F OIL Box Method: The box method does the exact same multiplications as our standard FOIL method, but gives it in a graphic organizer.

Be careful of positive and negatives.

Combine like terms of boxes to finish.

Exp 1: (x + 2) (x + 1)

Exp 2: (y + 3) (y - 4)

Exp 3: (a – 5 ) (a – 7 )

2

BINOMIAL #2

BIN

OM

IAL

#1 F

first terms

Oouter terms

Iinner terms

Llast

terms

Exp 4: (3x + 2) (x + 4)

Exp 5: (5b + 9) (b - 4)

Exp 6: (2n -7) (3n + 3)

Exp 7: (2x - 5) (2x - 5) Exp 8: (8r2 – 2r) (5r + 4)

Exp 9: (2x + 5y) (7y – 3x)

3

POLYNOMIALS: FOIL BOX METHOD Part 2

BINOMIAL TIMES TRINOMIAL: One More Column for 3rd term in trinomial

Example 1: (a + 3) (a2 + 7a + 6)

Example 3: (y - 5) (4y2 – 3y + 2)

Example 5: (x - 6) (x2 – 7x - 8 )

Example 2: (4x + 9) (2x2 – 5x + 3)

Example 4: (2b + 1) (b2 – 5b + 4)

Example 6: (3b2 – 4b) (2b2 – b + 7)

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Quadratic Equations

Main Idea/QuestionsNotes

StandardForm

All quadratic equations are written in the form: Graph When graphed, a quadratic equation creates a U-shaped curve called a ______________

Types of Parabolas

Using your graphing calculator, sketch the following:y= x2 + 2x – 5 y= -x2 + 3x +7

If ‘a’ is __________________________, then the parabola opens ____________, like a smile If ‘a’ is _________________________, then the parabola opens ____________, like a frown

Axis of Symmetry Formula for Axis of Symmetry:

Vertex

When the vertex is the ___________________ it is called a _______________ When the vertex is the ___________________ it is called a _______________ Use the _________________________, x, and solve for y to get the _______________

Example 1y= x2 + 8x + 15Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

26

Example 2y= -x2 - 5x + 10Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

You Try!#1y= 2x2 + 16x + 39

Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

#2y= x2 – 1 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

#3y= -x2 + 4x – 4 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

#4y= x2 + 3 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

#5y= 2x2 + 8x Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

27

#6y= 3x2 - 6x + 5 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

#7y= -x2 + 3x + 12 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

Graphing Quadratics PracticePart 1: Fill in the Blank1. A quadratic function is a function that can be written in the standard form: y = __________.2. Every quadratic function has a U-shaped graph called a __________________________.3. If the leading coefficient a is positive, the parabola ______________________________.4. If the leading coefficient a is negative, the parabola _____________________________.5. The _______________ is the lowest point of a parabola that opens ____.6. The _______________ is the highest point of a parabola that opens ____. 6. The line passing through the vertex that divides the parabola into two symmetric parts is called the ____________________________________________.7. Solutions of quadratic functions can also be called the __________.

Part 2: Draw a graph of the following functions and fill in all aspects of the graph.

#8y= x2 – 10 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

#9y= -2x2 + 4x – 9 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:28

#10y= x2 + 4x – 1 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

#11y= -2x2 + 8x – 8 Axis of Symmetry: __________ Vertex: ________________ Min/Max? ______________Sketch:

29

30

Summary of Quadratic Transformation Rules

31

F(x) + c moves the parent graph _____ c units F(x) - c moves the parent graph _____ c units F(x + c) moves the parent graph _____ c units F(x – c) moves the parent graph _____ c units -(f(x)) _______ the parent graph ___________________(over the ___________________________________) f(-x) _______ the parent graph ___________________(over the ___________________________________) a(f(x)) _______ or ________ the parent graph

If |a| > 1, then the graph _______ If 0 < |a| < 1, then the graph _________

Quadratic GraphsIn your graphing calculator, graph the function y = x2 (the quadratic parent function). Then graph each function below. Compare the new graph to the parent graph and write your observations about the location of the vertex, the overall shape, and the slope of the sides of the new graph in the blanks at the right.

Part A: The Effect of a

1. y=4 x2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

2.y= 1

4x2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

3. y=−4 x2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

32

4.y=−1

4x2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

Part B: The Effect of h

5. y=( x+2)2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

6. y=( x−4 )2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

33

7. y=−( x+5)2

Vertex: ______________________________________________

Shape Change or Shift Change? : ________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:____________________________________________

8. y=−( x−6 )2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

Part C: The Effect of k

9. y=x2+1

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

34

10. y=x2−2

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

11. y=−x2+7

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

12. y=−x2−10

Vertex: ______________________________________________

Shape Change or Shift Change? : _________________________

What was the change? _________________________________

Domain: _____________________________________________

Range:_______________________________________________

35

36

Solving Quadratics Equations GraphicallyThe solutions of a quadratic function are called ___________, _____________ or ____________. They occur when the graph ___________________________. 1. 2.

3. 4.

37

5. 6.

38

39

Types of Zeros Investigation

Given the following quadratic functions, find the zeros:

1. x2 - 49 = 0 2. x2 + 16 = 0 3. x2 = 0

Discuss with your partner the following questions:

1. When can you expect 2 solutions in a quadratic equation?2. When can you expect 1 solution in a quadratic equation? 3. When can you expect a quadratic to have no solutions?

Recall from the previous lesson, the quadratic formula:

x=−b±√b2−4ac2a

This part of the quadratic formula helps us to determine how many solutions a quadratic will have:

b2−4 ac

It is called the Discriminant.

Because the quadratic formula contains a square root, we can determine the number of solutions based on the discriminant.

If the discriminant is negative, how many solutions will the quadratic have?

If the discriminant is positive, how many solutions will the quadratic have?

If the discriminant is zero, how many solutions will the quadratic have?

40

Given the following quadratics, use the discriminant to determine how many solutions it will have.

1. x2 – 6x + 11 = 2__________________ 2. 3x2 + 5x = 12__________________

3. 3x2 + 48 = 0 __________________ 4. x2 – 27 = 0__________________

5. x2 + x + 1 = 0__________________ 6. x2 + 4x -1 = 0__________________

Given the following graphs of quadratic functions: a) determine the sign of the discriminant and b) whether the solutions are real or non-real.

41

Discriminant PracticeDirections: Find the discriminant and determine the number of solution each equation has.

42

43

44

45

46

47

48