PRIOR KNOWLEDGE: PROPERTIES OF LINEAR GRAPHS

WHAT ARE THE BASIC PROPERTIES OF A LINEAR GRAPH?

Introduction to Quadratic Graphs

However, not everything can be described using a linear

(straight line) graph.

So, We Know About the Main Properties of

Linear Graphs

Let’s Begin... There is a mythical creature called a “Walkasaurs” The table provided shows how “Walkasaurs” height changes with

time

Time (years) Height (metres)

0 1

1 2

2 4

3 7

4

5

6

7

Points to Ponder... Do you notice a pattern in the rate of growth of the

walkasaurus? Is the change in height each year the same?

(a constant number) Can you complete the table

for the remaining years? If I draw this graph will it be

a straight line(linear) ? Why or why not?

Time (years) Height (metres)

0 1

1 2

2 4

3 7

4

5

6

7

Drawing the graphComplete the table and plot the graph.

Time (years) Height (metres)

0 1

1 2

2 4

3 7

4

5

6

7

The Graph

2 4 6 8 10 12 14 16 18 200

5

20

15

10

25

30

35

40

45

Time (years)

Heig

ht

(metr

es)

Finding the Pattern

Time (years)

Height (metre

s) 1st change2nd

change0 1

1

1 2 12

2 4 13

3 7 14

4 11

The first change is not a constant number, as is the case in a linear graph, however the 2nd change is a constant, this is one of the properties of a quadratic graph.

Motor CyclistThe image below shows a motor cycle jumping a ramp. What “shape” is the path that the motor cycle follows?

The graph is curved, lets look at it in some more detail.. Is this the graph of a quadratic? Your Turn..See Handout

1.7 Pg. 5

Finding the PatternDistance travelled

(m)Height (m)

1st Change 2nd

Change0 0

+ 3.6

2 3.6 – 0.8+ 2.8

4 6.4 – 0.8+ 2

6 8.4 – 0.8+ 1.2

8 9.6 – 0.8+ 0.4

10 10 – 0.8– 0.4

12 9.6 – 0.8– 1.2

14 8.4 – 0.8– 2

16 6.4 – 0.8– 2.8

18 3.6 – 0.8– 3.6

20 0

Note: The second differences (or changes) are constant, therefore the graph is Quadratic

Speed(km/h)

180 240 300 360 420 480 540 600

Lift (net upward force)(Newtons)

11340 45360 102060 181440 283500 408240 555660 725760

For a given wing area the lift of an aeroplane is proportionalto the square of its speed. The table below shows the lift of aBoeing 747 jet airline at various speeds.

(a) Is the pattern of lifts quadratic? Give a reason for your answer.(b) Sketch the graph to show how the lift increases with speed.

A Boeing 747 weighs 46000 Newtons at takeoff . (c) Estimate how fast the plane must travel to get enough lift to take flight. (d) Explain why bigger planes need longer runways.

Aeroplane Lift Off

Speed(km/h)

180 240 300 360 420 480 540 600

Lift (net upward force)(Newtons)

11340 45360 102060 181440 283500 408240 55566072576

0

1st Change 34020 56700 79380 102060 124740 147420 170100

2nd Change 22680 22680 22680 22680 22680 22680

Speed (km/h)

Lift

(N

)

Because the second differences are constant, the pattern is quadratic.

See Geogebra File

Height 3 3.5 4 4.5 5

Distance 1.5 2.375 3.1 3.675 4.1

Angry Birds!!

Table of Values:

Angry Birds!!

Height 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

Distance 1.5 2.37

5 3.1 3.675 4.1 4.37

5

Height 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

Distance 1.5 2.37

5 3.1 3.675 4.1 4.37

5 4.5 4.475 4.3 3.97

5 3.5 2.875 2.1 1.17

5

Table of Values:

Angry Birds!!

See Geogebra file

Your turn.. See handoutGrowing Squares Pattern. Draw the next two patterns of growing squares.

Create a List of the Properties of Quadratic Graphs

1. They are curved.

2. The 1st change is not constant, but the 2nd change is constant

3. They can occupy all 4 quadrants of the plane

Introduction to Cubic Graphs

Cubic Graphs

As previously discussed, not every thing can be described by a straight line, nor can everything be described by a “ ” or “ ” shaped curve.

Lets take a look at the shape of a roller coaster.

It looks like 2 quadratics stuck together. But does it have the properties of a quadratic, i.e. The second differences will be constant?

Initial height = 0 m

Bird JourneySee animated power point on bird graph

Looking at the DataThe distance the bird travelled and its change in height relative to its starting position is given in the table below:

If we were to graph this data, what shape would the graph be?

Distance Travelled

(m)2 3 4 5 6 7 8

Change in height (m) 12 10 0 – 12 – 20 – 18 0

Looking at the Change in the DataDistance

Travelled (m) 2 3 4 5 6 7 8

Change in height (m) 12 10 0 –

12–

20–

18 0

1st Change – 2 – 10

– 12 – 8 2 18

2nd Change – 8 – 2 4 10 16

3rd Change 6 6 6 6

First change not a constant, so graph will not be LINEAR

Second change not a constant, so graph will not be QUADRATIC

Third change is a constant, this means the graph is a CUBIC

Graph of Bird’s Journey

Change in h

eig

ht(

m)

[Rela

tive t

o s

tart

ing p

osi

tion]

Distance travelled (m)

[Relative to starting position]

(2,12)

H

(3,10)

(4,0)

(5,–12)

(7,–18)

(6,–20)

(8,0)

For a cube with edge lengths of 1 unit,the perimeter of the base is 4 units,the surface area is 6 square units And the volume is 1 cubic unit.

What would the values be for a block withedge lengths of 2 units or 3 units or 34 units or n units?

Make tables for perimeter, for surface area and for volume as the edge lengths of the block increase.

Examine the tables to predict the shape of the graph for each of the three relationships.

Explain your predictions. Make the graphs for perimeter vs. edge length,surface area vs. edge length and volume vs. edge length and compare them with your predictions.

Using a Cube to Investigate Cubic FunctionsVertex

FaceEdge

1 unit

RECOGNIZE AND DESCRIBE AN EXPONENTIAL PATTERN.USE AN EXPONENTIAL PATTERN TO PREDICT A FUTURE

EVENT.COMPARE EXPONENTIAL AND LOGISTIC GROWTH.

Introducing Exponential Functions

Recognising an Exponential Pattern

A sequence of numbers has an exponential pattern when each successive number increases (or decreases) by the same percent.

Here are some examples of exponential patterns: Growth of a bacteria culture Growth of a mouse population during a mouse plague Decrease in the atmospheric pressure with increasing height Decrease in the amount of a drug in your bloodstream

Recognising an Exponential Pattern

Describe the pattern for the volumes of consecutive chambers in the shell of a chambered nautilus.

Solution: It helps to organize the data in a table.

Chamber 1 2 3 4 5 6 7

Volume (cm3) 0.836 0.889 0.945 1.005 1.068 1.135 1.207

Begin by checking the differences of consecutive volumes.

Source: Larson Texts

Recognising an Exponential Pattern

Begin by checking the differences of consecutive volumes to conclude that the pattern is not linear or Quadratic. Then find the ratios of consecutive volumes.

0 5 10 15 20 25 300.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Chamber 1 2 3 4 5 6 7

Volume (cm3) 0.836 0.889 0.945 1.005 1.068 1.135 1.207

Checking the Ratios

0.889 0.945 1.0051.063 1.063 1.063

0.836 0.889 0.9451.068 1.135 1.207

1.063 1.063 1.0631.005 1.068 1.135

The volume of each chamber is about 6.3% greater than the volume of the previous chamber. So, the pattern is exponential.

Notice the difference between linear and exponential patterns. With linear patterns, successive numbers increase or decrease by the same amount.

With exponential patterns, successive numbers increase or decrease by the same ratio.

Chamber 1 2 3 4 5 6 7

Volume (cm3) 0.836 0.889 0.945 1.005 1.068 1.135 1.207

Your Turn: See Handout Algae Bloom

Who Will Do Better?You and your friend have both been offered a job on a construction site.Both of you will have to work 28 consecutive days to finish the project.

Your friend is offered €25,000 per week. (for 4 weeks)

You negotiate your contact as follows:You can pay me 2 cent for the first day, 4 cent for the second day, 8 cent for the third day, and so on, just double my pay each day for 28 days.Who has negotiated the better deal?

End of Week 1

Time (days) Money (Cents)

0 2

1 4

2 8

3 16

4 32

5 64

6 128

7 256

Total: 510 cents (€5.10)

So at the end of week 1, You have earned €5.10, but your friend has earned €25,000. It would seem your friend has secured the better deal !

Table for the First 10 DaysView Handout

Time (days) Money (Cents)

0 2

1 4

2 8

3 16

4 32

5 64

6 128

7 256

8 512

9 1024

10 2048

But...What Will Happen After 28 Days?

Your final days pay will be €5,368,709.12Not bad for one days work!

Time (days) Money (Cents)

21 4,194,304

22 8,388,608

23 16,777,216

24 33,554,432

25 67,108,864

26 134,217,728

27 268,435.456

28 536,870,912

Both Graphs the Same but the Scales are Different

Tripling my pay

Tripling my payDoubling my pay

Doubling my pay

Exponential Graphs: Equation

xy abFinal

AmountStarting Value

Growth Factor

Intervals of time

Table for the First 10 Days

Time (days)

Money (Cents) Pattern

0 2 2 x 20 = 21

1 4 2 x 21 = 22

2 8 2 x 22 = 23

3 16 2 x 23 = 24

4 32 2 x 24 = 25

27 268,435,456 2x227 = 228

28 536,870,912 2x228 = 229

xy abCan you identify how the variables in the

above formula relate to the values in the

table?

View handout

The Power of Exponential Functions

Identifying Graphs..Your turnBelow are 4 sections of 4 different graphs, using the data provided, identify each type of graph, and give a reason for your answer.

Graph 1

Graph 2

Graph 3

Graph 4

Conclusion If a graph is Linear, the first change is

constant If a graph is quadratic, the second change is

constant If a graph is a cubic, the third change is

constant If a graph is exponential, successive

numbers increase or decrease by the same ratio.