Straight Lines Objectives: E Grade Plot the graphs of straight lines such as x = 3 and y = 4 Complete a table of values for equations such as y = 3x + 1 and draw the graph D Grade Solve problems involving graphs, such as finding where the line y = x +3 crosses the line y = 2 C Grade Recognise the equations of straight line graphs such as y = -3x + 1 Find the gradients of straight line graphs Prior knowledge: Plot co-ordinates in all four quadrants

Complete the table for the equation y = x The Gradient of a straight line Straight Lines x -3 -1 1 3 y   1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y -3 -1 1 3 x Complete the table for the equation y = 2x x x -3 -1 1 3 y   -6 -2 2 6 x x Complete the table for the equation y = 3x x -3 -1 1 3 y   -9 -3 3 9 Complete the table for the equation y = -x x -3 -1 1 3 y   3 1 -1 -3

What do you notice about these straight lines? 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y y = x y = 2x y = 3x y = -x x They are not parallel - they have different gradients

We look at the gradient more closely: 5 4 3 2 1 y = x Straight Lines We look at the gradient more closely: 5 4 3 2 1   y = x if x = 1 then y = 1 if x = 2 then y = 2 So we say : x “every time we go across 1 we go up 1” x 0 1 2 3 4 5 6

“every time we go across 1 we go up 2” x 0 1 2 3 4 5 6 Straight Lines   5 4 3 2 1   y = 2x x if x = 0 then y = 0 if x = 1 then y = 2 if x = 2 then y = 4 x So we say : “every time we go across 1 we go up 2” x 0 1 2 3 4 5 6

“every time we go across 1 we go up 3” x 0 1 2 3 4 5 6 Straight Lines x   5 4 3 2 1   y = 3x if x = 0 then y = 0 if x = 1 then y = 3 x if x = 2 then y = 6 So we say : “every time we go across 1 we go up 3” x 0 1 2 3 4 5 6

“every time we go across 1 we go down 1” x Straight Lines y = -x 2 1 -1 -2   if x = 0 then y = 0 if x = 1 then y = -1 if x = 2 then y = -2 x 0 1 2 3 4 5 6 So we say : x “every time we go across 1 we go down 1” x negative gradient A negative coefficient for x positive gradient A positive coefficient for x e.g. y = -x, y = -2x, y = -3x e.g. y = x, y = 2x, y = 3x

Straight Lines To summarise for the gradient of a line: The coefficient of x tells us the gradient of a straight line (how steep it is) A gradient of 1 “every time we go across 1 we go up 1” y = x y = 2x A gradient of 2 “every time we go across 1 we go up 2” Steeper than a gradient of 1 y = 3x A gradient of 3 “every time we go across 1 we go up 3” Steeper than a gradient of 2 y = -x A gradient of -1 “every time we go across 1 we go down 1” A negative gradient

Straight Lines To summarise for any straight line: For any equation in the form y = mx + c The variable m can be + or - m is the gradient c is the y-intercept

Straight Lines Finding the gradient for the line that passes through two pairs of coordinates: The gradient is the same at every point on the straight line. To find the gradient between two points find how much it has gone up and compare this with how much it has gone across. x How much up (y-direction) difference in y x How much across (x-direction) difference in x gradient =

Find the gradient for the line that passes through (2,1) and (5,7) Straight Lines Find the gradient for the line that passes through (2,1) and (5,7) 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y difference in y 7 – 1 = 6 difference in x 5 – 2 = 3 x 6 3 gradient = = 2 In general terms we can say (2,1) is (x1,y1) and (5,7) is (x2,y2) x Therefore: Gradient = y2- y1 x2 - x1

Straight Lines Now do these: = 3 = = -1 = =1 12 - 0 4 - 0 1 4 - 0 0 - 5 5 - 0 = -1 7 - 5 7 - 3 = 1 2 -4 - 5 -2 - 4 =1 1 2

Worksheet 1 Complete the table for the equation y = x Straight Lines x -3 -1 1 3 y   1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y Complete the table for the equation y = 2x x -3 -1 1 3 y   Complete the table for the equation y = 3x x -3 -1 1 3 y   Complete the table for the equation y = -x x -3 -1 1 3 y