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Piecewise Functions 6148 Views

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Description:

Don’t even think about starting up your combo lemonade/fro-yo stand without first making sure you understand piecewise functions. You’re just asking for your stand to... fall down.

Language:

English Language

Subjects:

Algebra / Functions

Tags:

Common Core Standards:

CCSS.Math.Content.8.F.B.5

Transcript

00:04
Piecewise Functions, a la Shmoop. John Lemon and Yoko Froyo run an unusually
00:11
successful and longstanding lemonade and yogurt stand in a highly populated metropolitan area.
00:16
It’s the end of a long day, and time to chart the daily earnings.
00:21
From 6 AM to noon, John runs the stand.
00:24
In order to attract people to the stand, he plays acoustic guitar and sings soothing,

00:28
classic rock songs at no extra charge.
00:34
He makes about 8 dollars an hour. Not bad for a lemonade and yogurt stand.
00:42
John’s earnings can be represented by the equation f of x equals 8 “x,”…
00:47
…where 8 dollars an hour times “x,”… the number of hours worked since 6 AM… will
00:54
give us the monetary total.
00:57
At noon, John leaves to work on his music, and Yoko takes over the stand until 6 PM.
01:01
To attract people to the stand, she does various performance art pieces at no extra charge.
01:08
Unfortunately, she doesn’t have as many fans as John. She makes about 5 dollars an
01:13
hour.
01:15
Yoko’s earnings can be represented by the equation g of y equals 5 “y,” where y
01:22
is the number of hours worked since noon.
01:26
How do we combine these two equations into one earnings chart? The answer lies in a piecewise
01:31
function. We can create a comprehensive piecewise function
01:42
that takes into account both John and Yoko’s total earnings over the course of the day.
01:46
Here’s what the function looks like…
01:48
F of “x” equals 8 “x” when “x” is greater than or equal to zero, and less
01:52
than or equal to 6 since John works from 6 AM till noon...
01:57
…AND 8 times 6… that’s John’s 48 dollars from his shift… plus 5 times… x minus
02:03
6…
02:03
…when “x” is greater than 6 and less than or equal to 12, since the stand closes
02:08
at 6 pm, 12 hours after 6 AM.
02:12
“x” is equal to the number of hours worked since 6 AM, which is why the second half of
02:17
the function…
02:18
…which represents Yoko’s earnings…
02:20
…contains “x” minus 6.
02:22
Her shift doesn’t start UNTIL noon, so the six hours before the switch don’t apply.
02:25
And of course, F of “x” is the total earnings of the day.
02:30
And now everything can be shown on one chart. All we do is graph the function.
02:34
Both pieces are line segments, so two points per piece are sufficient to graph the function:
02:41
For the first line segment for when “x” is greater than or equal to 0 or less than
02:46
or equal to 6…
02:47
…we can plug in x as 0, to get (0,0) as one endpoint.
02:54
To get the other endpoint of the first segment, we can plug in x as 6 to get (6, 48).
03:03
For the second line segment for when “x” is greater than 6 and less than or equal to
03:12
12…
03:12
…we can plug in "x" as 6 to get (6,48) to see where the second piece picks up from the
03:25
first.
03:27
To get the end point of the last piece, we can plug in 12 into the 8 times 6 plus 5 times…
03:34
x minus 6…
03:35
…to get 48 plus 5 times 6… 5 times 6 is 30, so 48 plus 30 equals 78.
03:45
So our endpoint for the second line segment is (12,78).
03:49
Perfect, we can graph it now.
03:52
Piecewise functions are just big functions made out of smaller pieces of other functions.
03:57
John loves them so much, he even has a song about them.
Full Transcript