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
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11
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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13
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
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)
17
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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
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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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20
Translations
• Shifting of a graph vertically and/or horizontally
• Size does not change
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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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22
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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23
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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24
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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25
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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27
-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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28
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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29
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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31
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
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- 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