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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

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.

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