29/01/2013SYSTEMS OF INEQUALITIES CHAPTER 3.2 PAGE 149 - 152

SOLVING SYSTEMS OF INEQUALITIES BY GRAPHING

TERMINOLOGY/KEYWORDS

Systems of InequalitiesBoundaryDashed LineSolid Line

LEARNING OBJECTIVES

1. Solving System of Inequalities by Graphing - Intersecting Regions.

2. Solving System of Inequalities by Graphing - Separate Regions.

3. Real World Problems – Write and use.

4. Find Vertices formed by graphs of systems of Inequalities.

DRAWING OF LINEAR INEQUALITY GRAPHS

Classwork3.2Pages 149- 152

Exercises 4(a, b), 7, 9, 10, 13, 14, 18, 19, 20, 23, 27(a, b, c), 47, 51 to 54.

Homework

Practice Worksheet

See website and Email

PRACTICE

SOLVING SYSTEMS OF LINEAR INEQUALITIES

1. We show the solution to a system of linear inequalities by graphing them.

a) This process is easier if we put the inequalities into Slope-Intercept Form, y = mx + b.

SOLVING SYSTEMS OF LINEAR INEQUALITIES

3. The solution also includes points not on the line, so you need to shade the region of the graph:

a) above the line for ‘y >’ or ‘y ’.b) below the line for ‘y <’ or ‘y ≤’.

SOLVING SYSTEMS OF LINEAR INEQUALITIES

Example: a: 3x + 4y > - 4

b: x + 2y < 2

Put in Slope-Intercept Form: ) 3 4 4

4 3 4

31

4

a x y

y x

y x

) 2 2

2 2

11

2

b x y

y x

y x

SOLVING SYSTEMS OF LINEAR INEQUALITIES

a:

dotted

shade above

b:

dotted

shade below

Graph each line, make dotted or solid and shade the correct area.

Example, continued:

3: 1

4 a y x 1

: 12

b y x

SOLVING SYSTEMS OF LINEAR INEQUALITIES

a: 3x + 4y > - 4

3: 1

4 a y x

SOLVING SYSTEMS OF LINEAR INEQUALITIES

a: 3x + 4y > - 4

b: x + 2y < 2

3: 1

4 a y x

1: 1

2 b y x

SOLVING SYSTEMS OF LINEAR INEQUALITIES

a: 3x + 4y > - 4

b: x + 2y < 2

The area between the green arrows is the region of overlap and thus the solution.

F(X) = X

2)( xxf

xxf )(

WE WILL NOW SEE HOW CERTAIN TRANSFORMATIONS (OPERATIONS) OF A FUNCTION CHANGE ITS GRAPH. THIS WILL GIVE US A BETTER IDEA OF HOW TO QUICKLY SKETCH THE GRAPH OF CERTAIN FUNCTIONS. THE TRANSFORMATIONS ARE (1) TRANSLATIONS, (2) REFLECTIONS, AND (3) STRETCHING/DILATION

0 = x23 = x2 + 3

VERTICAL TRANSLATIONS

f(x) = x2

f(x) +

01 = x2 +12 = x2+2

x

yy

VERTICAL TRANSLATIONS

f(x) = x2

f(x) +

0 = x2-1 = x2 -1-2 =x2 - 20-3 = x2 -3

x

yy

Adding c to f(x) moves the graph up by c units if c is positive, down if c is negative

HORIZONTAL TRANSLATIONS

f(x) = x2

f(x + 0) = (x+0)2f(x+1)=(x+1)2f(x+2) =(x+2)2 f(x+3) = (x+3)2

x

yy

HORIZONTAL TRANSLATIONS

f(x) = x2

f(x – 0) = (x-0)2f(x-1)=(x-1)2f(x-2) =(x-2)2 f(x-3) = (x-3)2

x

yy

Changing a function from f(x) to f(x-d) will move the graph d units to the right.

Changing a function from f(x) to f(x+d) will move the graph d units to the left.

COMBINING TRANSLATIONS

If f(x) = x2, graph f(x-2) +3:

f(x) = x2f(x-2)=(x-2)2f(x-2) +3 =(x-2)2 +3

x

yy

Parent Functions

Example 2 : Describe and graph Translations :Describe the translation in y = /x/ + 2

The graph of y = /x/ + 2 is a translation of the graph of y = /x/ up 2 units.Example 3 : Describe and graph Reflections:Describe the reflection in y = -x2.Then graph the function.

QUIZ 2

PG 110 1. 2A and 2B2.Describe the

Term Translation.3.Pg 113 –

Question 1 - 4

Example 3 : Describe and graph Reflections:Describe the reflection in y = .Then graph the function. 

The graph of y = is a reflection of the graph y = in the x- axis.

QUIZ 2

PG 110 1. Describe what is

a reflection and what is a reflection line.

2.Pg 113 Question 5 and 6.

Example 4 : Describe and graph DilationsDescribe the dilation in y = 4x .Then graph the function.

The graph of y = 4x is a dilation of y = x .The slope of the graph of y = 4x is steeper than that of the graph of y = x

Example 5 : Real – World Question - Identify TransformationsLanscaping:Etham is going to add a brick walkway around the perimeter of his vegetable garden. The area of the walkway can be represented by the function f(x) =4 (x+2.5)2 - 25.Describe the transformation in the function. Then graph the function.

The graph of f(x) =4 (x+2.5)2 - 25 is a combination of transformations of the parent graph f(x) = x2.

Determine how each transformation affects the parent graph.f(x) =4 (x+2.5)2 - 25+2.5 translates f(x) = x2 left 2.5 units -2.5 translates f(x) = x2 down 25 units4 stretches f(x) = x2 vertically

GROUP WORKIn groups of 4 look at the magic Equation and design your own Equation and draw graphs for your function.

Identify and describe the function.

RECALL “TRANSFORMING”

cdxaxf 2)()(

a = adjusting shape (compress, stretch or reflect)

c = moving up/down

d = moving left/right

Note: a ,c ,d R

Remember f(x) means – function with variable x

Lesson Quiz: Part I

Identify the parent function for g from its function rule. Then graph gby using grapher on your laptop and describe what transformation of the function it represents.

1. g(x) = x + 7

linear;translation up 7 units

Lesson Quiz: Part II

Identify the parent function for g from its function rule. Then graph g and describe what transformation of the parent function it represents.

2. g(x) = x2 – 7

quadratic;translation down 7 units

Check It Out!

The cost of playing an online video game depends on the number of months for which the online service is used. Graph the relationship from number of months to cost, and identify which parent function best describes the data. Then use the graph to estimate the cost of 5 months of online service.

The linear graph indicates that the cost for 5 months of online service is $72.

Step 1 Graph the relation.

Graph the points given in the table. Draw a smooth line through them to help you see the shape.

Step 2 Identify the parent function.

The graph of the data set resembles the shape of a linear parent function ƒ(x) = x.Step 3 Estimate the cost for 5 months of online service.

Check It Out! Example 3 Continued

Lesson Quiz: Part I

Identify the parent function for g from its function rule. Then graph g on your laptop and describe what transformation of the parent function it represents.

1. g(x) = x + 7

linear;translation up 7 units

Lesson Quiz: Part II

Identify the parent function for g from its function rule. Then graph g on your laptop and describe what transformation of the parent function it represents.

2. g(x) = x2 – 7

quadratic;translation down 6 units

Lesson Quiz: Part III3. Stacy earns $7.50 per hour. Graph the relationship

from hours to amount earned and identify which parent function best describes it. Then use the graph to estimate how many hours it would take Stacy to earn $60.

linear: 8 hr