08/01/2013Linear Relations and

FunctionsChapter 2 .7 Pg 109

Parent Functions and

Transformations

Terminology/Keywords

Terminology :Family of graphs, parent graph, parent function, constant function, identity function, quadratic function,

translation, reflection, line of reflection , dilation

Parent Functions

Learning Objectives

1.Identify a function given the graph.2. Describe and graph translations.3. Describe and graph Reflections.4.Identify Transformations.5.Real Life Problems.

Warm Up!Draw the following graphs

PRACTICE

Classwork

Pages 113 – 116 Exercises 1-8,9,10 – 1333 to 38 Bonus: 44 ,49 to 51

Homework

Page 116Exercises: 50.51

Practice Worksheet

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Similar to the way that numbers are classified into sets based on common characteristics, functions can be classified into families of functions. The parent function is the simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function.

The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

The identity function f(x) = x

The quadratic function

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

Parent Functions

Recall “Transforming”

cdxaxf 2)()(

a = adjusting shape (compress, stretch or reflect)

c = moving up/downd = moving left/rightNote: a ,c ,d R

Remember f(x) means – function with variable x

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

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