Introduction to Quadratic Graphs PRIOR KNOWLEDGE PROPERTIES OF

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Introduction to Quadratic Graphs PRIOR KNOWLEDGE: PROPERTIES OF LINEAR GRAPHS WHAT ARE THE BASIC

Introduction to Quadratic Graphs PRIOR KNOWLEDGE: PROPERTIES OF LINEAR GRAPHS WHAT ARE THE BASIC PROPERTIES OF A LINEAR GRAPH?

So, We Know About the Main Properties of Linear Graphs However, not everything can

So, We Know About the Main Properties of Linear Graphs However, not everything can be described using a linear (straight line) graph.

Let’s Begin. . . � There is a mythical creature called a “Walkasaurs” �

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

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) Time (years) � Can you complete the table 0 for the remaining years? 1 � If I draw this graph will it be a straight line(linear) ? 2 � Why or why not? 3 4 5 6 7 Height (metres) 1 2 4 7

Drawing the graph Complete the table and plot the graph. Time (years) Height (metres)

Drawing the graph Complete the table and plot the graph. Time (years) Height (metres) 0 1 1 2 2 4 3 7 4 5 6 7

The Graph 45 40 Height (metres) 35 30 25 20 15 10 5 0

The Graph 45 40 Height (metres) 35 30 25 20 15 10 5 0 2 4 6 8 10 12 Time (years) 14 16 18 20

Finding the Pattern Time (years) Height (metres) 0 1 1 st change 2 nd

Finding the Pattern Time (years) Height (metres) 0 1 1 st change 2 nd change 1 1 2 2 4 1 3 3 7 1 4 4 11 The first change is not a constant number, as is the case in a linear graph, however the 2 nd change is a constant, this is one of the properties of a quadratic graph.

Motor Cyclist The image below shows a motor cycle jumping a ramp. What “shape”

Motor Cyclist The 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

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 Pattern Distance travelled (m) Height (m) 0 0 2 3. 6 4

Finding the Pattern Distance travelled (m) Height (m) 0 0 2 3. 6 4 6 8. 4 8 9. 6 10 10 12 9. 6 14 8. 4 16 6. 4 18 3. 6 20 0 1 st Change + 3. 6 + 2. 8 +2 + 1. 2 + 0. 4 – 1. 2 – 2. 8 – 3. 6 2 nd Change – 0. 8 – 0. 8 Note: The second differences (or changes) are constant, therefore the graph is Quadratic

Aeroplane Lift Off For a given wing area the lift of an aeroplane is

Aeroplane Lift Off For a given wing area the lift of an aeroplane is proportional to the square of its speed. The table below shows the lift of a Boeing 747 jet airline at various speeds. Speed (km/h) Lift (net upward force) (Newtons) (a) (b) 180 240 300 360 420 480 540 600 11340 45360 102060 181440 283500 408240 555660 725760 Is the pattern of lifts quadratic? Give a reason for your answer. Sketch the graph to show 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.

Speed (km/h) Lift (net upward force) (Newtons) 180 240 300 360 420 480 540

Speed (km/h) Lift (net upward force) (Newtons) 180 240 300 360 420 480 540 600 11340 45360 102060 181440 283500 408240 555660 725760 1 st Change 2 nd Change 34020 56700 22680 79380 22680 102060 22680 124740 22680 147420 22680 170100 22680 Lift (N) Because the second differences are constant, the pattern is quadratic. See Geogebra File Speed (km/h)

Angry Birds!! Height Distance 3 3. 5 4 4. 5 5 1. 5 2.

Angry Birds!! Height Distance 3 3. 5 4 4. 5 5 1. 5 2. 375 3. 1 3. 675 4. 1

Angry Birds!! Table of Values: Height Distance 3 3. 5 4 4. 5 5

Angry Birds!! Table of Values: Height Distance 3 3. 5 4 4. 5 5 5. 5 1. 5 2. 375 3. 1 3. 675 4. 1 4. 375 6 6. 5 7 7. 5 8 8. 5 9 9. 5

Angry Birds!! See Geogebra file Table of Values: Height Distance 3 3. 5 4

Angry Birds!! See Geogebra file Table of Values: Height Distance 3 3. 5 4 4. 5 5 5. 5 6 6. 5 7 7. 5 8 8. 5 9 9. 5 1. 5 2. 375 3. 1 3. 675 4. 1 4. 375 4. 475 4. 3 3. 975 3. 5 2. 875 2. 1 1. 175

Your turn. . See handout Growing Squares Pattern. Draw the next two patterns of

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

Create a List of the Properties of Quadratic Graphs 1. They are curved. 2.

Create a List of the Properties of Quadratic Graphs 1. They are curved. 2. The 1 st change is not constant, but the 2 nd change is constant 3. They can occupy all 4 quadrants of the plane

Introduction to Cubic Graphs

Introduction to Cubic Graphs

Cubic Graphs As previously discussed, not every thing can be described by a straight

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?

Bird Journey See animated power point on bird graph Initial height = 0 m

Bird Journey See animated power point on bird graph Initial height = 0 m

Looking at the Data The distance the bird travelled and its change in height

Looking at the Data The distance the bird travelled and its change in height relative to its starting position is given in the table below: Distance Travelled (m) 2 3 4 5 6 7 8 Change in height (m) 12 10 0 – 12 – 20 – 18 0 If we were to graph this data, what shape would the graph be?

Looking at the Change in the Data Distance Travelled (m) 2 3 4 Change

Looking at the Change in the Data Distance Travelled (m) 2 3 4 Change in height (m) 12 10 0 1 st Change 2 nd Change 3 rd Change – 2 5 – 2 6 7 8 – 12 – 20 – 18 – 10 – 12 – 8 6 – 8 4 6 2 10 6 18 16 6 0 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 [Relative to starting position] Change in height(m) (2, 12) (3,

Graph of Bird’s Journey [Relative to starting position] Change in height(m) (2, 12) (3, 10) H (4, 0) (8, 0) (5, – 12) (7, – 18) (6, – 20) Distance travelled (m) [Relative to starting position]

Using a Cube to Investigate Cubic Functions For a cube with edge lengths of

Using a Cube to Investigate Cubic Functions 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 with edge lengths of 2 units or 34 units or n units? Vertex Edge Face 1 unit 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.

Introducing Exponential Functions RECOGNIZE AND DESCRIBE AN EXPONENTIAL PATTERN. USE AN EXPONENTIAL PATTERN TO

Introducing Exponential Functions RECOGNIZE AND DESCRIBE AN EXPONENTIAL PATTERN. USE AN EXPONENTIAL PATTERN TO PREDICT A FUTURE EVENT. COMPARE EXPONENTIAL AND LOGISTIC GROWTH.

Recognising an Exponential Pattern � A sequence of numbers has an exponential pattern when

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

Recognising an Exponential Pattern Describe the pattern for the volumes of consecutive chambers in the shell of a chambered nautilus. Source: Larson Texts Solution: It helps to organize the data in a table. Chamber 1 2 3 4 5 6 7 Volume (cm 3) 0. 836 0. 889 0. 945 1. 005 1. 068 1. 135 1. 207 Begin by checking the differences of consecutive volumes.

Recognising an Exponential Pattern Chamber 1 2 3 4 5 6 7 Volume (cm

Recognising an Exponential Pattern Chamber 1 2 3 4 5 6 7 Volume (cm 3) 0. 836 0. 889 0. 945 1. 005 1. 068 1. 135 1. 207 4. 0 3. 5 3. 0 2. 5 2. 0 1. 5 1. 0 0. 5 0. 0 0 5 10 15 20 25 30 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.

Checking the Ratios Chamber 1 2 3 4 5 6 7 Volume (cm 3)

Checking the Ratios Chamber 1 2 3 4 5 6 7 Volume (cm 3) 0. 836 0. 889 0. 945 1. 005 1. 068 1. 135 1. 207 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.

Your Turn: See Handout Algae Bloom

Your Turn: See Handout Algae Bloom

Who Will Do Better? You and your friend have both been offered a job

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

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 Days View Handout Time (days) Money (Cents) 0 2

Table for the First 10 Days View 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? Time (days) Money (Cents) 21

But. . . What Will Happen After 28 Days? 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 Your final days pay will be € 5, 368, 709. 12 Not bad for one days work!

Both Graphs the Same but the Scales are Different Tripling my pay Doubling my

Both Graphs the Same but the Scales are Different Tripling my pay Doubling my pay

Exponential Graphs: Equation Intervals of time Final Amount Starting Value Growth Factor

Exponential Graphs: Equation Intervals of time Final Amount Starting Value Growth Factor

Table for the First 10 Days View handout Time (days) Money (Cents) Pattern 0

Table for the First 10 Days View handout Time (days) Money (Cents) Pattern 0 2 2 x 20 = 2 1 1 4 2 x 21 = 2 2 2 8 2 x 22 = 23 3 16 2 x 23 = 24 4 32 2 x 24 = 2 5 27 268, 435, 456 2 x 227 = 228 28 536, 870, 912 2 x 228 = 229 Can you identify how the variables in the above formula relate to the values in the table?

The Power of Exponential Functions

The Power of Exponential Functions

Identifying Graphs. . Your turn Below are 4 sections of 4 different graphs, using

Identifying Graphs. . Your turn Below 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 q If a graph is Linear, the first change is constant q If

Conclusion q If a graph is Linear, the first change is constant q If a graph is quadratic, the second change is constant q If a graph is a cubic, the third change is constant q If a graph is exponential, successive numbers increase or decrease by the same ratio.