1. Straight Lines Objectives: B GradeExplore the gradients of parallel straight line graphs A GradeExplore the gradients of perpendicular straight line graphs Prior knowledge:Recognise the equations of straight line graphs and find the gradients of straight line graphs Rearrange equations to make a variable the subject

2. Straight Lines Find the equation of a line parallel to 2x + y = 4 that crosses the y axis at - 3 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 First, rearrange the equation in the form y = mx + c y = - 2x + 4 Identify the gradient In this case - 2 Any line with the same gradient is parallel The line that has the same gradient and has a y-intercept of - 3 is: y = - 2x - 3 Parallel Lines

3. Straight Lines Finding the equations of the lines parallel to the following equations and that pass through the coordinates given: 1.y = 3x(0,5) 2.y = - x(0,2) 3.y = 2x + 4(0, - 3) 4.y = 3x - 2(0, - 1) 5.y = - 4x - 4(0, 3) 6.2x – y = 2(0, - 7) 7.2y – 6x = 4(0, - 4) 8.2x + 4y = 4(0, 5) 9. y = 2x + 1 (0, 2) 10. y = - x - 4 (0, 1) 1313 y = 3x + 5 y = - x + 2 y = 2x - 3 y = 3x - 1 y = - 4x + 3 y = 2x - 7 y = 3x - 4 1212 y = - x + 5 y = 6x + 2 y = - x + 1

4. Straight Lines Perpendicular Lines Perpendicular means “at right angles to” For the line y = x draw a line that is perpendicular to it. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = x y = - x Notice how the gradient is now negative. The gradient of a perpendicular line is always the opposite sign.

5. Straight Lines For the line y = 2x draw a line that is perpendicular to it. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = 2x y = - ½x

6. Straight Lines For the line y = 3x draw a line that is perpendicular to it. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = 3x y = - x 1313

7. Straight Lines For the line y = 4x draw a line that is perpendicular to it. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = 4x y = - x 1414

8. Straight Lines Look at these equations together: y = x Perpendicular line y = - x y = 2x y = - ½x y = 3x y = - x 1313 y = 4x y = - x 1414 To summarise: The gradient of a perpendicular line is the negative reciprocal of the gradient of the original line.

9. Straight Lines What is the equation of the line perpendicular to y = 2x + 3 that goes through (0,5) The gradient of this line is 2, so the gradient of the line perpendicular to it is - ½ The line crosses the y-axis (the line x = 0) at (0,5), so the y-intercept is 5 The equation is therefore: y = - ½ x + 5

10. Straight Lines Now do these: y = - x + 15 y = x + 15 y = - x - 15 y = x - 15