New Zealand
Level 7 - NCEA Level 2

# Describing functions

Lesson

A function is like an equation that relates an input to an output. We commonly express functions in the form:

$f(x)=$f(x)=...

We can think of it like a machine that has an input and an output, like you can see below. We take a value, $x$x, apply a rule (i.e. put it through the function machine) and get an output value, $f(x)$f(x). The output value can also be referred to as "$f$f of $x$x".

To interpret information from a graph we need to understand some particular features of the graph

• What do the axes represent?
• Is the line increasing, decreasing, or flat?
• What does the steepness of individual sections tell us?

Combining these features together allows us to make interpretive statements and conclusions about the context.

##### Example 1

What are our 3 questions?

## What do the axes represent?

The horizontal axis (or $x$x-axis) is time, measured in seconds. As time passes we follow the graph from left to right. This will tell us a story about what is happening. The object starts at $0$0 seconds and the graph stops tracking the object at point $F$F at $25$25 seconds.

The vertical axis (or $y$y-axis) is speed, measured in m/s. As we move higher up the axis the speed gets faster. So the higher the line goes the faster the object is moving.

## Is the line increasing, decreasing or flat?

• for the parts where the line is increasing (going up, like segments $OA$OA and $DE$DE) this means that the object is speeding up
• for the parts where the line is decreasing (going down, like segment $BC$BC) this means that the object is slowing down
• for the flat sections $AB$AB and $CD$CD, we can see that the object had constant speed. During $AB$AB the object stayed at $10$10 m/s and during $CD$CD it was not moving at $0$0 m/s. During $EF$EF the object stayed at $5$5 m/s.

## What does the steepness of individual sections tell us?

How steep a section is will tell us how quickly the speed is changing. We can tell from the graph that $BC$BC is steeper that $AO$AO. This means that the speed is changing more quickly during section $BC$BC than $AO$AO.

## Why look at the context?

Once we have thought about the three context questions, we are now well prepared to answer anything about the graph or the object. It is really important that you always take some time to understand the context before trying to jump in an answer questions. The understanding that you get from doing that will help to ensure you get the questions correct!

##### Example 2

This graph may look more complicated, but by asking our $3$3 questions we can get a picture of what is happening.

## What do the axes represent?

The horizontal or $x$x-axis is time, measured in seconds. As time passes we move from left to right. This graph tells the story of an object starting at $A$A, and $200$200 seconds later finishing at $F$F, (back where it started).

The vertical or $y$y-axis is distance, measured in metres. As the graph gets higher up the $y$y-axis, the object is further away from the beginning. This graph tells the story of an object that travels $150$150 m away from the starting point, but then returns to the starting point.

## Is the line increasing, decreasing or flat?

For lines that are increasing, the object is moving away from the starting point. For lines that are decreasing the object is moving back towards the starting point.

## What does the steepness of individual sections tell us?

The steeper a line, the more distance traveled in a short amount of time. This means that steeper lines indicate a faster moving object. Flat lines tell us that the object is not moving at all.

Now that we have an understanding we can actually create the story of this object.

• Starting at $A$A, the object takes $20$20 seconds to travel $40$40 m. We can work out the speed of this section, as $40$40 m/$20$20 s which is $2$2 m/s. This gets us to $B$B
• The object then remains stationary (doesn't move) for $40$40 seconds. This gets us to $C$C.
• The object then travels $110$110 meters in $60$60 seconds at a variable speed. (Variable because it is not straight, so we cannot work out the speed here, but we could work out an average speed if we wanted). This gets us to $D$D.
• The object again remains stationary for another $40$40 seconds. This gets us to $E$E.
• The object then returns to the initial position, taking $40$40 seconds to travel $150$150 m. The speed of this section is $150$150 m/$40$40 s = $3.75$3.75 m/s.

We could get even more creative with our story, and use less numeric detail.

A butterfly set off on his lunch time feeding trip. Heading to flower $B$B (a blue flower $40$40 meters away) at a constant speed. Sitting on this flower for $40$40 seconds the butterfly has a lovely drink. Taking off to flower $D$D, the butterfly starts of slow, then speeds up before landing on the flower $110$110 meters away. Again sitting on this flower the butterfly drinks for $60$60 seconds and then briskly returns home to avoid capture by the Blue Wren in the area.

Try making your own story with a graph to describe it- perhaps a car chase or a roller coaster ride.

Football fans might want to check out this fun interactive for how graphs can describe scenarios in a football match. (Note: it will only work on a Flash enabled device, ... sorry )

Remember!

There are different types of data. Make sure you're familiar with them by reading through Sorting Your Info.

#### Worked Examples

##### Question 1

The graph shows Charlie's speed while he is competing in a bike race. Which situation corresponds to the graph?

1. Charlie starts off by increasing his pace gradually and then stops as he trips over a stick, but then recovers and is able to increase his speed steadily until the end of the race

A

Charlie starts off at a constant speed and then increases his speed at a steady rate

B

Charlie hits his maximum speed about halfway through the race

C

Charlie increases his speed at a constant rate throughout

D

Charlie starts off by increasing his pace gradually and then stops as he trips over a stick, but then recovers and is able to increase his speed steadily until the end of the race

A

Charlie starts off at a constant speed and then increases his speed at a steady rate

B

Charlie hits his maximum speed about halfway through the race

C

Charlie increases his speed at a constant rate throughout

D

##### Question 2

The book "The Life and Times of A Circle" is sold at a particular bookstore for $\$1010 each.

1. Is the revenue generated, as a function of the number of copies sold, continuous or discrete data?

Continuous

A

Discrete

B

Continuous

A

Discrete

B
2. Plot the revenue generated against the number of copies sold.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems