friday: announcements today ends the 2 nd week of this 5 week grading period passing back quiz #2...

FRIDAY: Announcements• TODAY ends the 2nd week of this 5

week grading period

• Passing back Quiz #2 today

• Tuesday is Quiz #3

• Thursday is your first UNIT TEST (60%)

Calendar Key Links

• Parent Function Packet answers

• Factoring Packets (Answer Keys)

• All notes for this unit so far

• Quiz Correction Forms (Precalc Tab)

Quiz Corrections• Correct any problems you missed

(except bonus)

• Due on test day!!

• Show all work for the reworked problems. Don’t just give a new answer!

• Graded for accuracy based on– How many were wrong– How many did you fix– How many were correct

Graph: f(x) = 0

Constant

Graph: f(x) = x

Linear

Graph: f(x) = x2

Quadratic

Graph: f(x) = 1

x

Reciprocal

Graph: f(x) = x

Square Root

Factoring Cubes

3 3a b 2 2( )( )a b a ab b

3 3a b 2 2( )( )a b a ab b

Greatest Integer Function(GIF)and

Transformations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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Objectives• I can find functional values of the

Greatest Integer Function (GIF)• I can graph the Greatest Integer

Function• I can identify characteristics of the GIF• I can recognize the order of

transformations

NEW function page

GREATEST INTEGER FUNCTION

(GIF)The greatest integer function is a

piece-wise defined function.

The GIF is like the bill for your cell phone, but in reverse. If you talk for 4 ½ minutes you get charged for 4 minutes.


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Greatest Integer Function• f(x) = [x] or sometimes f(x) = [[x]]

• This generates the greatest integer less than or equal to the value of x

• Examples: [2.7] = 2

• [-3.6] = -4

• [1/3] = 0

Remembering GIF

• Use a number line!

• ALWAYS round Down

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Start with a dark circle on the origin. The dark horizontal line is 1 unit long. It has an open circle on the right.

Greatest Integer FunctionThe domain of this function is all real numbers.

The range is all integers (Z)

Would the absolute value function be even or odd or neither?

[[ ]]f x x

Transformations

• Review from Algebra-2

• Types

• - Translations (left, right, up, down)

• - Reflections (x-axis, y-axis)

• - Size Changes (dilations)

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18

Transformation Rules• Equation How to obtain the graph

For (c > 0)

• y = f(x) + c Shift graph y = f(x) up c units

• y = f(x) - c Shift graph y = f(x) down c units

• y = f(x – c) Shift graph y = f(x) right c units

• y = f(x + c) Shift graph y = f(x) left c units


19

Transformation Rules• Equation How to obtain the graph

• y = -f(x) (c > 0) Reflect graph y = f(x) over x-axis

• y = f(-x) (c > 0) Reflect graph y = f(x) over y-axis

• y = af(x) (a > 1) Stretch graph y = f(x) vertically by

factor of a

• y = af(x) (0 < a < 1) Compress graph y = f(x) vertically by factor of a

• Multiply y-coordinates of y = f(x) by a


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Translations

• Shifting of a graph vertically and/or horizontally

• Size does not change


21

f (x)

f (x) + c

+c

f (x) – c-c

If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units.

Vertical Shifts

If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units.

x

y


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h(x) = |x| – 4

Example: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4.

f (x) = |x|

x

y

-4 4

4

-4

8

g(x) = |x| + 3


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Graphing Utility: Sketch the graphs given by 2,y x 2 1, andy x 2 3.y x

–5 5

4

–4

2+1 y x

2y x

2 3y x


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x

y

y = f (x) y = f (x – c)

+c

y = f (x + c)

-c

If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units.

Horizontal Shifts

If c is a positive real number, then the graph of f (x + c) is the graph of y = f

(x) shifted to the left

c units.


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f (x) = x3

h(x) = (x + 4)3

Example: Use the graph of f (x) = x3 to graph

g (x) = (x – 2)3 and h(x) = (x + 4)3 .

x

y

-4 4

4

g(x) = (x – 2)3


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Graphing Utility: Sketch the graphs given by 2,y x 2( 3) , andy x 2( 1) .y x

–5 6

7

–1

2( 3)y x

2y x

2( 1)y x


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-4

y

4

x-4

x

y4

Example: Graph the function using the graph of .

First make a vertical shift 4 units downward.

Then a horizontal shift 5 units left.

45 xyxy

(0, 0)(4, 2)

(0, – 4)(4, –2)

xy

4 xy

45 xy

(– 5, –4) (–1, –2)


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y = f (–x) y = f (x)

y = –f (x)

The graph of a function may be a reflection of the graph of a basic function.

The graph of the function y = f (–

x) is the graph of y = f (x) reflected in the y-axis.

The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x-axis.

x

y


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The graphs of many functions are transformations of the graphs of very basic functions.

The graph of y = –x2 is the reflection of the graph of y = x2 in the x-axis.

Example: The graph of y = x2 + 3 is the graph of y = x2 shifted upward three units. This is a vertical shift.

x

y

-4 4

4

-4

-8

8

y = –x2

y = x2 + 3

y = x2


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x

y

4

4y = x2

y = – (x + 3)2

Example: Graph y = –(x + 3)2 using the graph of y = x2.

First reflect the graph in the x-axis.

Then shift the graph three units to the left.

x

y

– 4 4

4

-4

y = – x2

(–3, 0)


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Vertical Stretching and Compressing

If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c.

If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) compressed vertically by c.

Example: y = 2x2 is the graph of y = x2 stretched vertically by 2.

– 4x

y

4

4

y = x2

is the graph of y = x2

compressed vertically by .

2

4

1xy

4

1

2

4

1xy

y = 2x2


32

- 4x

y

4

4

y = |x|

y = |2x|

Horizontal Stretching and Shrinking

If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by c.

If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by c.

Example: y = |2x| is the graph of y = |x| shrunk horizontally by 2.

xy2

1

is the

graph of y = |x| stretched

horizontally by .

xy2

1

2

1


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Graphing Utility: Sketch the graphs given by 3,y x 3, d0 an1 y x 3.1

10y x

–5 5

5

–5

310y x

3y x

3110

y x


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- 4

4

4

8

x

y

Example: Graph using the graph of y = x3.3)1(2

1 3 xy

3)1(2

1 3 xyStep 4:

- 4

4

4

8

x

y

Step 1: y = x3

Step 2: y = (x + 1)3

3)1(2

1 xyStep 3:

Graph y = x3 and do one transformation at a

time.


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function blue The 2

function red The

xy

xy

function red The 3

functiongreen The

xy

xy

Homework

• WS 1-6

friday: announcements today ends the 2 nd week of this 5 week grading period passing back quiz #2...

Documents

graph of y

fx c shift graph y

fx c shift graph y

fx c shift graph y

graph of f x c

graph of f x c

fx right c units y

f x f x c c f x ccif