# Positive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin.

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1 1. Introduction To describe position we need a fixed reference (start) point and a way to measure direction and distance. In Mathematics we use Cartesian coordinates, named after the Mathematician and Philosopher René Descartes ( ). The fixed reference point is called the Origin. The horizontal axis is labelled x. The vertical axis is labelled y. Coordinates are listed as (x, y). The origin is (0, 0) Positive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin. A is at (3, 2). From the origin, 3 to the right and 2 up. B is at (1, 5) C is at ( 4, 3). From the origin, 4 to the left and 3 down. D is at ( 6, 4) E is at (2, 5) Plot these F (5, 6) G ( 3, 2) H (4, 2) I ( 1, 1) J (0, 4) K ( 1, 0) Page 1 of 28

2 What do all these points have in common? Describe the line that would join the points. x = 2 Any point on this line has an x-value of 2 They all meet the criterion x = 2 The line has the equation x = Page 2 of 28

3 Exercise 1a What are the equations of the following lines? Draw the lines x = 6, x = 2, x = 3, x = 5 What word could you use to describe the direction of these lines? So any vertical line has an equation of the form x = a number The y-axis has equation x = Page 3 of 28

4 What do all these points have in common? Describe the line that would join the points. y = 4 Any point on this line has a y-value of 4. They all meet the criterion y = 4 The line has the equation y = Page 4 of 28

5 Exercise 1b What are the equations of the following lines? Draw the lines y = 6, y = 2, y = 3, y = 5 What word could you use to describe these lines? So any horizontal line has an equation of the form y = a number The x-axis has equation y = Page 5 of 28

6 2. Drawing a straight line from its equation Cartesian coordinates and straight lines booklet Think of a straight road that you are walking along. You might describe it as being flat (horizontal), or having an upward slope as you go up a hill. In mathematics we describe lines by using an equation. This is the rule that links the x and y values of any point on the line. Consider the line y = 2x (remember that 2x = 2 x) To draw a line we need a minimum of two points. We will use three points for confirmation. Let s set up a table: x-value Why 0, 1, and 2? Because they are y-value nice, easy numbers. We substitute these x-values into the equation y = 2x to work out the related y -value. y = 2x x = 0 y = 2 0 = 0 x = 1 y = 2 1 = 2 x = 2 y = 2 2 = 4 x-value y-value So our coordinates are (0, 0) (1, 2) (2, 4) Page 6 of 28

7 Now we plot these and draw the line through them. Cartesian coordinates and straight lines booklet Any point on this line will meet the criterion y= 2x y = 2x Check this for yourself: x = 3 y = 6 x = 2 y = 4 y= 3x Page 7 of 28

8 How about the line y = 3x 1? y = 3x 1 x = 0 y = 1 x = 1 y = 2 x = 2 y = 5 x-value y-value Remember BIDMAS? We Multiply before Subtracting = 0 1 = = 3 1 = = 6 1 = 5 So our coordinates are (0, 1) (1, 2) (2, 5) Page 8 of 28

9 Exercise 2 a. Draw the line y = 3x 5 y = 3x 5 x = 0 y = x = 1 y = x = 2 y = x-value y-value The coordinates are (0, ) (1, ) (2, ) Page 9 of 28

10 b. Draw the line y = x + 4 y = x + 4 x = 0 x = 1 x = 2 x-value y-value The coordinates are (0, ) (1, ) (2, ) Page 10 of 28

11 c. Draw the line y = 6 5x y = 6 5x x = 0 x = 1 x = 2 x-value y-value Page 11 of 28

12 d. Draw the line y = ½x + 3 y = ½x + 3 x = 0 x = 2 x = 4 x-value y-value Page 12 of 28

13 e. Draw the line x + y = 6 x + y = 6 x = = 6 x = 1 1+? = 6 x = 2 2+? = 6 x-value y-value This equation may look a bit different from previous ones but don t be put off. If in doubt, substitute values and have a look. What number plus 2 gives 6? Page 13 of 28

14 f. Draw the line y = 2x + 3 y = 2x + 3 x = 0 y = x = 1 y = 2(1) + 3 = x = 2 y = x-value y-value Page 14 of 28

15 3. Gradient The gradient of a line is the ratio of the vertical change in distance (height or rise) to the distance travelled horizontally (run). In road signs this is often expressed as a percentage. In mathematics we usually express a gradient as a fraction. gradient = vertical change (y) horizontal change (x) = y x The Greek letter delta, Δ is often used to denote a change in a quantity. gradient = vertical change (y) horizontal change (x) = y x Page 15 of 28

16 Let s compare these two graphs: Cartesian coordinates and straight lines booklet y= x + 3 What effect does a negative coefficient of x have on a line? y= 2x Page 16 of 28

17 Let s compare these two graphs: Cartesian coordinates and straight lines booklet y= 2x 3 What do these two equations have in common? What word could you use to describe the direction of these lines? Hint: railway tracks! y= 2x Page 17 of 28

18 Let s compare these two graphs: Cartesian coordinates and straight lines booklet If we pick any point on the line then move 4 right and 3 up, we return to the line. gradient = vertical (y) horizontal (x) = 3 4 gradient = vertical (y) horizontal(x) = Page 18 of 28

19 gradient = vertical (y) horizontal (x) = 4 1 = 4 Exercise 3a From the starting points on this grid, mark on two further points for each line, counting boxes, then draw the lines with these gradients: gradient A 3 3 up 1 right B up 2 right C down 3 right Page 19 of 28

20 If we are given the coordinates of two points we can calculate the gradient of the line segment joining them. Label the points (x 1, y 1 ) and (x 2, y 2 ). gradient = Δy Δx = y 2 y 1 x 2 x 1 Example (Check this calculation by counting along and up) (x 1, y 1 ) ( 1, 2 ) (x 2, y 2 ) ( 5, 4 ) gradient = y 2 y 1 x 2 x 1 = = 2 4 = Page 20 of 28

21 Does it matter which way we label (x 1, y 1 ) and (x 2, y 2 ). Let s check by reversing this last example: Cartesian coordinates and straight lines booklet (x 2, y 2 ) ( 1, 2 ) (x 1, y 1 ) ( 5, 4 ) gradient = y 2 y 1 x 2 x 1 = = 2 4 = 1 2 The gradient is the same. Exercise 3b Work out the gradient of the lines between these points: A (3, 2) and B (7, 6) C ( 3, 3) and D (9, 7) E ( 5, 2) and F( 3, 8) G (11, 2) and H (3, 2) Page 21 of 28

22 4. Working out an equation of a straight line Cartesian coordinates and straight lines booklet This is often expressed as y = mx + c It s important to identify the things that are different and the things that are the same. What about the equations we ve looked at? y = 3x 5 y = 6 5x x + y = 6 y = x + 4 y = 1 2 x + 3 y = 2x + 3 The coefficient of x and the number on its own can vary but these all fit the pattern of the straight line equation. Note: y = 2x is really y = 2x + 0 y = 3 is really y = 0x + 3 Memorise: y = mx + c m = gradient = Δy Δx = y 2 y 1 x 2 x 1 c = y intercept (where the line crosses the y-axis) Page 22 of 28

23 Example What is the equation of this line? Choose points on the line that are nice and easy to read. m = y 2 y 1 x 2 x 1 = 3 ( 3) 0 ( 4) = 6 4 = 3 2 The line crosses the y-axis at y = 3, therefore c = 3. y = 3 2 x Page 23 of 28

24 Example What is the equation of this line? Choose points on the line that are nice and easy to read. m = y 2 y 1 x 2 x 1 = = 4 6 = 2 3 The line crosses the y-axis at y = 1 therefore c = -1. y = 2 3 x Page 24 of 28

25 Exercise 4 Work out the equations of these lines: a. b. c Page 25 of 28

26 5. Quick method to draw a straight line Cartesian coordinates and straight lines booklet If asked to draw a line, given the equation, a quick way is to mark the y-intercept and then count along and up/down to get a second point. Example: Draw the line y = 5x + 4 Exercise 5 a. Draw the line y = 2x Page 26 of 28

27 b. y = 4x 3 c. y = 4 3 x 1 d. y = 2 x + 4 c. y = 3x Page 27 of 28

28 Solutions Exercise 1a x = 4 x = 0 Exercise 1b y = 0 y = 3 Exercise 2 a. (0, -5), (1, -2), (2, -1) b. (0, 4), (1, 5), (2, 6) c. (0, 6), (1, 1), (2, -4) d. (0, 3), (1, 3.5), (2, 4) e. (0, 6), (1, 5), (2, 4) f. (0, 3), (1, 1), (2, -1) Exercise 3b Gradient AB = 1 Gradient CD = 1/3 Gradient EF = 5 Gradient GH = -1/2 Exercise 4 a. y = 2x + 3 b. y = 1 2 x + 2 c. y = 3x Page 28 of 28

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