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

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Properties of Linear Graphs Line is straight Line can travel from left to right, or right to left Line can only be a maximum of 3 quadrants on the plane Rate of change (slope) is constant everywhere on the line

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

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Lets Begin... There is a mythical creature called a Walkasaurs The table provided shows how Walkasaurs height changes with time Time (years)Height (metres) 01 12 24 37 4 5 6 7

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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) 01 12 24 37 4 5 6 7

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Drawing the graph Complete the table and plot the graph. Time (years)Height (metres) 01 12 24 37 4 5 6 7

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The Graph 2468101214161820 0 5 15 10 25 30 35 40 45 Time (years) Height (metres)

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Finding the Pattern Time (years) Height (metres) 1 st change 2 nd change 01 1 12 1 2 24 1 3 37 1 4 411 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.

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Motor Cyclist The image below shows a motor cycle jumping a ramp. What shape is the path that the motor cycle follows?

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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

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Finding the Pattern Distance travelled (m) Height (m) 1 st Change 2 nd Change 00 + 3.6 23.6 – 0.8 + 2.8 46.4 – 0.8 + 2 68.4 – 0.8 + 1.2 89.6 – 0.8 + 0.4 10 – 0.8 – 0.4 129.6 – 0.8 – 1.2 148.4 – 0.8 – 2 166.4 – 0.8 – 2.8 183.6 – 0.8 – 3.6 200 Note: The second differences (or changes) are constant, therefore the graph is Quadratic

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Speed (km/h) 180240300360420480540600 Lift (net upward force) (Newtons) 1134045360102060181440283500408240555660725760 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. (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

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Speed (km/h) 180240300360420480540600 Lift (net upward force) (Newtons) 1134045360102060181440283500408240555660725760 1 st Change340205670079380102060124740147420170100 2 nd Change22680 Speed (km/h) Lift (N) Because the second differences are constant, the pattern is quadratic. See Geogebra File

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Height 33.544.55 Distance 1.52.3753.13.6754.1 Angry Birds!!

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Table of Values: Angry Birds!! Height 33.544.555.566.577.588.599.5 Distance 1.52.3753.13.6754.14.375

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Height 33.544.555.566.577.588.599.5 Distance 1.52.3753.13.6754.14.3754.54.4754.33.9753.52.8752.11.175 Table of Values: Angry Birds!! See Geogebra file

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

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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

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Introduction to Cubic Graphs

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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?

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

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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: If we were to graph this data, what shape would the graph be? Distance Travelled (m) 2345678 Change in height (m) 12100– 12– 20– 180

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Looking at the Change in the Data Distance Travelled (m) 2345678 Change in height (m) 12100– 12– 20– 180 1 st Change– 2– 10– 12– 8218 2 nd Change– 8– 241016 3 rd Change6666 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

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

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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 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 Functions Vertex Face Edge 1 unit

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

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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

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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 1234567 Volume (cm 3 ) 0.8360.8890.9451.0051.0681.1351.207 Begin by checking the differences of consecutive volumes. Source: Larson Texts

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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. Chamber 1234567 Volume (cm 3 ) 0.8360.8890.9451.0051.0681.1351.207

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Checking the Ratios 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 1234567 Volume (cm 3 ) 0.8360.8890.9451.0051.0681.1351.207

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Your Turn: See Handout Algae Bloom

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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?

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End of Week 1 Time (days)Money (Cents) 02 14 28 316 432 564 6128 7256 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 !

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Table for the First 10 Days View Handout Time (days)Money (Cents) 02 14 28 316 432 564 6128 7256 8512 91024 102048

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But...What Will Happen After 28 Days? Your final days pay will be 5,368,709.12 Not bad for one days work! Time (days)Money (Cents) 214,194,304 228,388,608 2316,777,216 2433,554,432 2567,108,864 26134,217,728 27268,435.456 28536,870,912

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

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

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Table for the first 10 days View Handout Time (days) Money (Cents) First chang e Second change Pattern 022x2 0 =2 1 142x2 1 =2 2 282x2 2 =2 3 3162x2 3 =2 4 4322x2 4 =2 5 564 6128 7256 8512 271024 2 4 8 16 32 64 12 8 25 6 51 2 1024 2 4 16 32 64 12 8 25 6 51 2 8

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Table for the First 10 Days Time (days) Money (Cents) Pattern 022 x 2 0 = 2 1 142 x 2 1 = 2 2 282 x 2 2 = 2 3 3162 x 2 3 = 2 4 4322 x 2 4 = 2 5 27268,435,4562x2 27 = 2 28 28536,870,9122x2 28 = 2 29 Can you identify how the variables in the above formula relate to the values in the table? View handout

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The Power of Exponential Functions

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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 1Graph 2Graph 3Graph 4

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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.

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