29/01/2013 Systems of Inequalities Chapter 3.2 PAGE 149 - 152 Solving Systems of inequalities BY GRAPHING

Terminology/Keywords Systems of Inequalities Boundary Dashed Line Solid Line

Solving System of Inequalities by Graphing - Intersecting Regions. Learning Objectives Solving System of Inequalities by Graphing - Intersecting Regions. Solving System of Inequalities by Graphing - Separate Regions. Real World Problems – Write and use. Find Vertices formed by graphs of systems of Inequalities.

Drawing of linear inequality graphs

Classwork Homework 3.2 Pages 149- 152 Practice Worksheet 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

Solving Systems of Linear Inequalities We show the solution to a system of linear inequalities by graphing them. This process is easier if we put the inequalities into Slope-Intercept Form, y = mx + b.

Solving Systems of Linear Inequalities The solution also includes points not on the line, so you need to shade the region of the graph: above the line for ‘y >’ or ‘y ’. 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:

Solving Systems of Linear Inequalities Example, continued: Graph each line, make dotted or solid and shade the correct area. a: dotted shade above b: dotted shade below

Solving Systems of Linear Inequalities a: 3x + 4y > - 4

Solving Systems of Linear Inequalities a: 3x + 4y > - 4 b: x + 2y < 2

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

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

Vertical Translations f(x) = x2 f(x) + y y 1 = x2 +1 2 = x2+2 3 = x2 + 3 0 = x2 x

Vertical Translations f(x) = x2 f(x) + y y -3 = x2 -3 0 = x2 -1 = x2 -1 -2 =x2 - 2 x 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 y y f(x+3) = (x+3)2 f(x+2) =(x+2)2 f(x + 0) = (x+0)2 f(x+1)=(x+1)2 x

Horizontal Translations f(x) = x2 y y f(x-3) = (x-3)2 f(x-2) =(x-2)2 f(x – 0) = (x-0)2 f(x-1)=(x-1)2 x 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: y y f(x-2)=(x-2)2 f(x) = x2 f(x-2) +3 =(x-2)2 +3 x

Parent Functions

Example 3 : Describe and graph Reflections: 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 2B 2.Describe the Term Translation. 3.Pg 113 – Question 1 - 4

Example 3 : Describe and graph Reflections: Describe the reflection in y = −𝑥 2 .Then graph the function.   The graph of y = −𝑥 2 is a reflection of the graph y = −𝑥 2 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 Dilations Describe 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 Transformations Lanscaping: 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. 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 units 4 stretches f(x) = x2 vertically The graph of f(x) =4 (x+2.5)2 - 25 is a combination of transformations of the parent graph f(x) = x2.

Group work In 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” 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.

Check It Out! Example 3 Continued 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. The linear graph indicates that the cost for 5 months of online service is $72.

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

linear: 8 hr Lesson Quiz: Part III 3. 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