Math with Michelle- Functions Trends & Periodicity

Math with Michelle- Functions Trends & Periodicity

Trends and Periodicity

We can predict the shape of some functions' graphs for large values of x because they describe events with a very clear trend.

Sample problem: Graph of trend of a girl's height

• Over how long was the study carried out?

• What was her height at birth? And after 20 years?

• What can we determine about her heigh in 10 years' time?

Sample problem: Graph of trend of a girl's height

• Her height was studied from the time she was born until she was 25 years old

• When she was born she was 50 cm long. At 20 years old she was 175 cm tall

• Her height will be the same as it is when she is 25 years old, 175 cm.

Radiator Temperature

• How long was it's temperature 25ºC?

• When was it turned off?

• How long did it take to cool down?

• If it is not turned on again, what will its temperature be in 3 hours?

Radiator Temperature

• It was at 25ºC for 2.5 hours

• It was turned off at 2.5 hours/ at half past 2 in the morning

• It took 1 hour to cool down

• It will be 10ºC in 3 hours if it is not turned on again

Pedro wants to sell his computer

He knows that since he has bought his computer it has lost some of its value each year.

Multiple Choice

How much did it cost when he bought it?

• 1000€

• 100€

• 800€

Multiple Choice

If he wants to sell it for half of what it cost him, when should he sell it?

• After 2 years

• After 3 years

• After 4 years

• After 5 years

• After 6 years

Can you observe any trend in this function?

Can you observe any trend in this function?

It tends towards 0

Fill in the Blanks

Describe the trend of the function depicted in this graph:

It tends towards

Fill in the Blanks

Describe the trend of the function depicted in this graph:

It tends towards

Periodicity

Periodic functions are those whose behaviour is repeated each time the independent variable covers a certain interval.

the length of this interval is called a period

Sample problem

• The graph shows the distance that separates Juan from his daughter, who is spinning around on a carousel.

Sample problem

• How long does it take her to complete a full spin? How many spins are shown?

• What is the maximum distance between Juan and his daughter? When do these maxima occur?

Sample problem

• We see that the distance increases from 2m to 12m and then decreases from 12m to 2m. That happens every 20 seconds, which is how long 1 spin lasts. It shows 4 spins.

• The maximum distance is at 12m and it occurs at 10s, 30s, 50s and 70s

Sample problem

• If the carousel continues to spin, what will the distance be in 90 seconds?

• Explain why the function is periodic and say what its period is.

Sample problem

• At 90 seconds the distance will be at 12m.

• It's a periodic function because its values are repeated every 20 seconds. The period is 20 seconds.

Tank of water: filling and emptying

A tank of water used for watering plants is filled in 4 minutes, then 2 minutes later it is emptied in 10 minutes, after which it is filled and emptied again.

Multiple Choice

What is the tank's capacity?

• 30 litres

• 25 litres

• 20 litres

Multiple Select

When is it full?

• Between 0-4 minutes

• Between 4-6 minutes

• Between 6-16 minutes

• Between 20-22 minutes

Continue the function from minute 32 to minute 48

Continue the function from minute 32 to minute 48

Multiple Choice

What is the function's period?

• 4 minutes

• 2 minutes

• 16 minutes

Distance of a comet from the sun over time

Fill in the Blanks

What is the period?

Complete the graph

to show the distance covered as the comet completes 3 orbits

Complete the graph

to show the distance covered as the comet completes 3 orbits

Continuity and Discontinuity

Continuous function

When you can plot its graph without lifting your pencil off the paper

Discontinuous function

When a function contains points at which it is not continuous

Example

• A mobile phone salesman is paid 600€ a month plus 10€ commission for each phone he sells. This is the graph of this function

• phones sold ---> monthly earnings

Example

• The number of phones sold can be 0, 1, 2, ... 10. It cannot be a fraction or decimal number.

• This means the points cannot be connected.

• The graph is discontinuous

Car park rates

The first 2 hours are free

Each further hour or fraction of an hour: €1.50

This is the graph of this function

parking time ---> cost

Multiple Choice

How much does it cost to park for 3 hours?

• €0

• €1.50

• €3

Multiple Choice

How much does it cost to park for 3 and a quarter hours?

• €1.50

• €3

• €4.50

Explain why the graph makes a jump at x=2, x=3, ...

Explain why the graph makes a jump at x=2, x=3, ...

Because if we park more than 2 hours we go from paying €0 to paying €1.50

When 3h is exceed there is another jump, because we now have to pay €3.

Multiple Choice

Is this function continuous or discontinuous?

• continuous

• discontinuous

A group of friends wants to buy Carmen a computer as a graduation present

This graph describes the amount of money each friend will have to contribute

Depending on how many friends join their money together

Why can't the points be connected?

A group of friends wants to buy Carmen a computer as a graduation present

Because the number of friends can be 1, 2, 3,... but it cannot be a fraction or decimal

Multiple Choice

Is the function continuous or discontinuous?

• continuous

• discontinuous

Fill in the Blanks

Explain the function's trend as the number of friends grows