Post on 22-Jan-2018
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Section 3-9Step Functions
Warm-upName the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2 + 3
Warm-upName the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2 + 3
2
Warm-upName the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2 + 3
2 3
Warm-upName the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2 + 3
2 3 4
Warm-upName the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2 + 3
2 3 4
0
Warm-upName the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2 + 3
2 3 4
0 -102
Warm-upName the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2 + 3
2 3 4
0 -102 3
Greatest-Integer:
Greatest-Integer: The greatest integer less than or equal to x
x⎢⎣ ⎥⎦
Greatest-Integer: The greatest integer less than or equal to x
x⎢⎣ ⎥⎦
Step Function:
Greatest-Integer: The greatest integer less than or equal to x
x⎢⎣ ⎥⎦
Step Function: A graph that looks like a series of steps, with each “step” being a horizontal line segment
Greatest-Integer: The greatest integer less than or equal to x
x⎢⎣ ⎥⎦
Step Function: A graph that looks like a series of steps, with each “step” being a horizontal line segment
*It is a function, so it must pass the Vertical-Line Test*
Greatest-Integer: The greatest integer less than or equal to x
x⎢⎣ ⎥⎦
Step Function: A graph that looks like a series of steps, with each “step” being a horizontal line segment
*It is a function, so it must pass the Vertical-Line Test*
*Each step will include one endpoint*
Example 1Simplify.
a. 4⎢⎣ ⎥⎦
b. −7 25
⎢⎣ ⎥⎦ c. 3.2⎢⎣ ⎥⎦
Example 1Simplify.
a. 4⎢⎣ ⎥⎦
b. −7 25
⎢⎣ ⎥⎦ c. 3.2⎢⎣ ⎥⎦
4
Example 1Simplify.
a. 4⎢⎣ ⎥⎦
b. −7 25
⎢⎣ ⎥⎦ c. 3.2⎢⎣ ⎥⎦
4 -8
Example 1Simplify.
a. 4⎢⎣ ⎥⎦
b. −7 25
⎢⎣ ⎥⎦ c. 3.2⎢⎣ ⎥⎦
4 -8 3
Greatest-Integer Function
Greatest-Integer Function
The function f where f (x ) = x⎢⎣ ⎥⎦
for all real numbers x.
Greatest-Integer Function
The function f where f (x ) = x⎢⎣ ⎥⎦
for all real numbers x.
*Also known as the rounding-down function*
Greatest-Integer Function
The function f where f (x ) = x⎢⎣ ⎥⎦
for all real numbers x.
*Also known as the rounding-down function*...because we’re rounding down
Example 2 Graph f (x ) = x⎢⎣ ⎥⎦ +1.
Example 2 Graph f (x ) = x⎢⎣ ⎥⎦ +1.
1. Set up a table
Example 2 Graph f (x ) = x⎢⎣ ⎥⎦ +1.
1. Set up a table
2. Determine the length of each interval
Example 2 Graph f (x ) = x⎢⎣ ⎥⎦ +1.
1. Set up a table
2. Determine the length of each interval
3. Choose an integer for one endpoint and the next integer for the other
Example 2 Graph f (x ) = x⎢⎣ ⎥⎦ +1.
1. Set up a table
2. Determine the length of each interval
3. Choose an integer for one endpoint and the next integer for the other
4. Determine which endpoint is included by testing a value in between
Example 2 Graph f (x ) = x⎢⎣ ⎥⎦ +1.
1. Set up a table
2. Determine the length of each interval
3. Choose an integer for one endpoint and the next integer for the other
4. Determine which endpoint is included by testing a value in between
5. Finish your table and plot your graph
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥ = 3⎢⎣ ⎥⎦
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥ = 3⎢⎣ ⎥⎦ = 3
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥ = 3⎢⎣ ⎥⎦ = 3
3 rolls
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥ = 3⎢⎣ ⎥⎦ = 3
78650
⎢
⎣⎢
⎥
⎦⎥
3 rolls
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥ = 3⎢⎣ ⎥⎦ = 3
78650
⎢
⎣⎢
⎥
⎦⎥ = 15.72⎢⎣ ⎥⎦
3 rolls
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥ = 3⎢⎣ ⎥⎦ = 3
78650
⎢
⎣⎢
⎥
⎦⎥ = 15.72⎢⎣ ⎥⎦ =15
3 rolls
Example 3Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786
pennies?
p
50
⎢
⎣⎢
⎥
⎦⎥
15050
⎢
⎣⎢
⎥
⎦⎥ = 3⎢⎣ ⎥⎦ = 3
78650
⎢
⎣⎢
⎥
⎦⎥ = 15.72⎢⎣ ⎥⎦ =15
3 rolls 15 rolls
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
−3 x − 2
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
−3 x − 2 1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
−3 x − 2 1.5 −1.5 1− (−3)⎢⎣ ⎥⎦ = −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
−3 x − 2 1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
−3 x − 2 1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
−3 x − 2 1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
−3 x − 2 1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 = −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
−2 < x ≤ −1
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
−2 < x ≤ −1 -1.5
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
−2 < x ≤ −1 -1.5
−1< x ≤ 0
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0
0 < x ≤ −1
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0
0 < x ≤ −1 1.5
= −4.5
Example 4 Graph f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦ .
x f (x ) =1.5 −1.5 1− x⎢⎣ ⎥⎦
1.5 −1.5 1− (−3)⎢⎣ ⎥⎦
1.5 −1.5 1− (−2)⎢⎣ ⎥⎦ = −3
1.5 −1.5 1− (−2.5)⎢⎣ ⎥⎦ = −3 −3 < x ≤ −2
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0
0 < x ≤ −1 1.5
= −4.5
Homework
Homework
p. 189 #1-24, skip 12, 16
“There ain’t no free lunches in this country. And don’t go spending your whole life commiserating that you got raw deals. You’ve got to say, ‘ I think that if I keep working at this and want it bad enough I can have it.’” - Lee Iacocca