Coordinates and Graphs Year 9

Note 1: Coordinates Coordinates require 2 references: 1.) Horizontal (written first) 2.) Vertical (written second) Coordinates can be used to identify a place of interest.

Note 1: Coordinates and Maps

Topographical Maps show many features including: heights (elevations) contours vegetation water and coastal features roads, railways, bridges and tunnels residential and historical landmarks

Note 1: Coordinates and Maps The caravan park at Orongo Bay has a grid reference Split the reference into 2 parts The first 3 digits (152) represent the horizontal scale The second 3 digits (569) represent the vertical scale

Note 1: Mathematical Coordinates In math, we use coordinates to represent a position on a whole number plane The coordinate of point A is _______ The coordinate of point B is _______ (1,3) (5,2) * Remember, the first coordinate gives the distance across (horizontal) and the second gives the distance up (vertical)

Note 1: Mathematical Coordinates The horizontal scale is the x-axis The vertical scale is the y-axis (x,y)

Note 1: Mathematical Coordinates The horizontal scale is the x-axis The vertical scale is the y-axis (x,y) IWB Ex pg Ex pg Ex pg IWB Ex pg Ex pg Ex pg

Note 2: Coordinates with integers We can extend the x and y axes to include negative numbers * Remember we always write x before y (0,0) is the point where the x-axis and the y-axis intersect. This point is called the origin. (x,y)

Note 2: Coordinates with integers IWB Ex pg IWB Ex pg

Note 3: Latitude & Longitude

Latitude – horizontal lines (run east –west) Longtitude – vertical lines (run north-south) e.g. Equator e.g. Prime Meridian Greenwich, London, England

Note 3: Latitude & Longitude

IWB Ex pg 431 IWB Ex pg 431

Note 4: Two-dimensional graphs Scatter Plots shows the relationship between the two quantities two axes, each with a different quantity and scale

Note 4: Two-dimensional graphs Scatter Plots This scatter plot shows the relationship between height and age 1.) Who is the youngest? 2.) Who is the tallest? 3.) If these are all children in a single family, who are the twins? 4.) What is the relationship that this graph shows between height and age? Cindy Aaron Bruce & Emma The older a person is, the taller they are (positive relationship)

Note 4: Two-dimensional graphs Scatter Plots This scatter plot shows the distances of some overseas destinations from Auckland airport, and the cost of the return ticket to each one. a.) Which destination is the furthest from Auckland? b.) Which destination is the most expensive to travel to? c.) How does the graph show that the fare to Sydney and Melbourne is the same? Seoul L.A. They are on the same horizontal line With a fare of $500

Note 4: Two-dimensional graphs Scatter Plots This scatter plot shows the distances of some overseas destinations from Auckland airport, and the cost of the return ticket to each one. d.) The distance to Adelaide is 3250 km and the fare is $799, add this to the graph. e.) What relationship does this graph show exists between distance and cost ? The further away destinations are more expensive to travel to (positive relationship)

Note 4: Two-dimensional graphs Line Graphs Show how one quantity changes as another one does two axes, each with a different quantity and scale

Note 4: Two-dimensional graphs Line Graphs This graph shows what happens to two plants after they were planted in a garden. One received fertilizer and the other did not. a.) What was their height when they were planted? b.) Which plant grew the fastest? c.) How does the graph show when the plants stop growing? 10 cm Plant A – it has a steeper slope (gradient) Plant A stopped after 16 days and plant B after 18 days – their graph becomes horizontal after day 16 & 18.

Note 4: Two-dimensional graphs Line Graphs Sue is adding water to a storage tank with a hose but doesn’t realize that there is a leak in the tank. When the tank looks full, she stops adding water and walks away. a.) What is the level in the tank when she starts filling it? b.) How long does it take to fill the tank? c.) At what level is there a hole in the tank? 0.4 m = ____ cm40 20 min 0.8 m = ____ cm 80 IWB Ex pg 435 Ex pg 440 IWB Ex pg 435 Ex pg 440

Note 5: Graphs & Coordinate Patterns An algebra rule that links x and y can be used to plot coordinates that produce a pattern on a graph e.g. y = 2x x ×-2 = -4 2×-1 = -2 2 × 0 = 0 2 × 1 = 2 2 × 2 = 4 2 × 3 = 6 Coordinates (-2,-4) (-1,-2) (0, 0 ) (1, 2 ) (2, 4 ) (3, 6 ) Once the coordinates are plotted the pattern is obviousLinear

Note 5: Graphs & Coordinate Patterns An algebra rule that links x and y can be used to plot coordinates that produce a pattern on a graph e.g. y = 3x – 1 x ×-2-1 = -7 3×-1-1 = -4 3× 0-1 = -1 3× 1-1 = 2 3× 2-1 = 5 3× 3-1 = 8 Coordinates (-2,-7) (-1,-4) (0, -1) (1, 2 ) (2, 5 ) (3, 8 ) Linear Pattern

Note 5: Graphs & Coordinate Patterns 3 × -1 = -3 3 × 0 = 0 3 × 1 = 3 3 × 2 = 6 3 × 3 = 9 (-1,-3) (0, 0) (1, 3 ) (2, 6 ) (3, 9 ) Linear Pattern IWB Ex pg IWB Ex pg

Starter -2 × -1 – 1 = 1 -2 × 0 – 1 = × 1 – 1 = × 2 – 1 = × 3 – 1 = -7 (-1, 1) (0, -1) (1,-3 ) (2, -5 ) (3, -7 ) Linear Pattern y = -2x × -2 – 1 = 3 (-2, 3) -2x – 1 = y

Note 6: Connecting Tables, Rules & Graphs Rule: W = 10 – 2n W = amount of water (L) n = number of minutes elapsed since the bucket was full 1.) What is the value of W when n=1? 2.) How much water is left in the bucket after 2 min.? 3.) How much water is in the bucket when it is full? 4.) How long does it take to empty the bucket? 8 Litres 6 Litres 10 Litres 5 minutes

Note 6: Connecting tables, rules & graphs C = 2n = 2 x

Note 6: Connecting tables, rules & graphs C = 2n + 30 At time 0, the cost is $30. (y-intercept) When C = $41, n = 5.5 minutes

IWB Ex pg 465 IWB Ex pg 465 Note 6: Connecting tables, rules & graphs $ 0 $ 1.5 $ 4.5 $ 6.0 $ 7.5 Does it make sense to extend the line below the x-axis? Yes, it shows a loss if they sell less than 4 apples

Note 7: Gradients The gradient of a line is a number that tells us how steep it is We measure gradients by describing how much the line rises for every 1 unit of sideways movement

Note 7: Gradients The gradient of a line is a number that tells us how steep it is This line has a gradient of 2 We can think of it as the fraction 2 1

Note 7: Gradients What are the gradients of these lines? rise run rise run 1111 = = 1 = rise run 2323

Note 7: Gradients rise run 1111 = = 2

Note 8: Negative Gradients rise run -2 1 = Write down the gradients of these lines -2 2 = -1 BETA Ex 27.6 pg 371 Ex 27.7 pg 372 BETA Ex 27.6 pg 371 Ex 27.7 pg 372

Note 9: y-intercept The y-intercept of a line is a number that tells us where it crosses the y-axis y-intercept gradient