Straight Line Graphs Straight Line Graphs 1 Sections

Straight Line Graphs

Straight Line Graphs 1) Sections Horizontal, Vertical and Diagonal Lines (Exercises) 2) y = mx + c (Exercises : Naming a Straight Line Sketching a Straight Line) 3) Plotting a Straight Line - Table Method (Exercises) 4) Plotting a Straight Line – X = 0, Y = 0 Method (Exercises) 5) Supporting Exercises Co-ordinates Negative Numbers Substitution

Naming horizontal and vertical lines y (x, y) (3, 4) 4 3 2 (3, 1) 1 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 x y = -2 -2 -3 (-4, -2) -4 (0, -2) -5 (-4, -2) (3, -5) x=3 Back to Main Page

Now try these lines y (x, y) (-2, 4) 4 3 -5 -4 -3 -2 -1 2 y=2 1 (-2, 1) 0 -1 1 2 3 4 5 x -2 -3 (-4, 2) -4 (0, 2) x = -2 -5 (-4, 2) (-2, -5) Back to Main Page

See (x, y) if you can name lines 1 to 5 y 4 x = -4 x=1 x=5 3 2 y=1 1 -5 -4 -3 -2 -1 0 1 2 3 1 4 5 x -1 -2 y = -4 5 -3 4 -4 -5 2 3 Back to Main Page

Diagonal Lines (x, y) y=x+1 y y=x 4 (-3, 3) (3, 3) 3 2 (-1, 1) -5 (1, 1) 1 -4 -3 (2, -2) -2 -1 0 -1 1 2 3 4 x 5 -2 (-3, -3) -3 (-4, -3) -4 (0, 1) -5 (2, 3) y = -x Back to Main Page

Now see if you can identify these diagonal lines y=x+1 y 4 3 y=x-1 3 y=-x-2 2 1 -5 -4 -3 -2 -1 0 -1 -2 1 2 3 4 x 5 y = -x + 2 -3 -4 1 2 -5 4 Back to Main Page

y = mx + c Every straight line can be written in this form. To do this the values for m and c must be found. c is known as the intercept y = mx + c m is known as the gradient Back to Main Page

Finding m and c y 8 Find the Value of c 7 This is the point at which the line crosses the y-axis. 6 5 4 So c = 3 3 2 1 – 7 – 6 – 5 – 4 – 3 – 2 – 1 -1 1 2 3 4 5 -2 -3 -4 -5 -6 yy == mx 2 x +3 +c 6 7 8 x Find the Value of m The gradient means the rate at which the line is climbing. Each time the lines moves 1 place to the right, it climbs up by 2 places. So m = 2 Back to Main Page

Finding m and c y 8 Find the Value of c 7 This is the point at which the line crosses the y-axis. So c = 2 6 5 yy == mx 2 x +3 +c 4 3 2 1 – 7 – 6 – 5 – 4 – 3 – 2 – 1 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6 7 8 x Find the Value of m The gradient means the rate at which the line is climbing. Each time the line moves 1 place to the right, it moves down by 1 place. So m = -1 Back to Main Page

Some Lines to Identify y Line 1 8 m =1 7 c= 5 Equation: 4 – 5 – 4 – 3 – 2 – 1 2 m =1 1 c= -1 -2 -3 -4 -5 -6 y=x+2 Line 2 3 – 7 – 6 2 6 1 2 3 4 5 6 7 8 -1 x Equation: y = x - 1 Line 3 m = -2 c= 1 Equation: y = -2 x + 1 Back to Main Page

Exercise y 5 8 Click for Answers 3 7 6 5 4 3 2 1 – 7 – 6 – 5 – 4 – 3 – 2 – 1 -1 1 2 3 4 5 6 7 8 -2 x 1) y=x-2 2) y = -x + 3 3) y = 2 x + 2 4) y = -2 x - 1 5) y = -2 x - 1 6) 2 -3 2 -4 1 -5 -6 4 Back to Main Page

Further Exercise Sketch the following graphs by using y=mx + c 1) y=x+4 6) y=1–x 2) y=x-2 3) y = 2 x + 1 7) 8) y = 3 – 2 x y = 3 x 4) y = 2 x – 3 9) 5) y = 3 x – 2 y=x+2 2 y=-x+1 2 10) Back to Main Page

The Table Method We can use an equation of a line to plot a graph by substituting values of x into it. Example y = 2 x + 1 x=0 y = 2(0) +1 y=1 x 0 1 2 x=1 y = 2(1) +1 y=3 y 1 3 5 x=2 y = 2(2) +1 y=5 Now you just have to plot the points on to a graph! Back to Main Page

The Table Method x y 0 1 1 3 2 4 3 5 2 1 -4 -3 0 -2 -1 1 2 3 4 -1 y = 2 x + 1 -2 -3 -4 Back to Main Page

The Table Method Use the table method to plot the following lines: 1) y=x+3 2) y = 2 x – 3 3) y=2–x 4) y = 3 – 2 x x 0 1 2 y Click to reveal plotted lines Back to Main Page

The Table Method 4 3 2 1 -4 -3 0 -2 -1 1 2 3 4 -1 -2 -3 1 3 -4 2 4 Click for further exercises Back to Main Page

Further Exercise Using the table method, plot the following graphs. 1) y = x + 2 7) y=1–x 2) y = x – 3 8) y = 1 – 2 x 2 3) y = 2 x + 4 4) y = 2 x – 3 9) y = 2 – 3 x 5) y = 3 x + 1 10) y=x+1 6) y = 3 x – 2 2 Back to Main Page

The x = 0, y = 0 Method This method is used when x and y are on the same side. Example: x + 2 y = 4 To draw a straight line we only need 2 points to join together. Back to Main Page

If we find the 2 points where the graph cuts the axes then we can plot the line. These points are where x = 0 (anywhere along the y axis) and y = 0 (anywhere along the x axis). Back to Main Page

y 8 7 6 5 This is where the graph cuts the x – axis (y=0) 4 This is where the graph cuts the y – axis (x=0) 3 2 1 -6 -5 -4 -3 -2 -1 - 1 1 2 3 4 5 6 7 8 x -2 -3 -4 -5 -6 Back to Main Page

By substituting these values into the equation we can find the other half of the co-ordinates. Back to Main Page

Example Question: Draw the graph of 2 x + y = 4 Solution x=0 y=0 2(0) + y = 4 2 x + 0 = 4 y=4 2 x = 4 x=2 1 st Co-ordinate = (0, 4) 2 nd Co-ordinate = (2, 0) Back to Main Page

So the graph will look like this. y 8 2 x + y = 4 7 6 5 4 3 2 1 – 7 – 6 – 5 – 4 – 3 – 2 – 1 -1 1 2 3 4 5 6 7 8 x -2 -3 -4 -5 -6 Back to Main Page

Exercise Plot the following graphs using the x=0, y=0 method. 1) x+y=5 2) x + 2 y = 2 3) 2 x + 3 y = 6 4) x + 3 y = 3 Click to reveal plotted lines Back to Main Page

y Answers 8 7 6 5 1. 3 x + 2 y = 6 4 2. x + 2 y = 2 2 3. 2 x + 3 y = 6 1 4. x - 3 y = 3 3 – 7 – 6 – 5 – 4 – 3 – 2 – 1 -1 1 2 3 4 5 6 7 8 x -2 -3 -4 Click for further exercises -5 -6 Back to Main Page

Exercise Using the x = 0, y = 0 method plot the following graphs: 1) x+y=4 6) x–y=3 2) 2 x + y = 2 7) 2 x – y = 2 3) x + 2 y = 2 8) 2 x – 3 y = 6 4) x + 3 y = 6 9) x + 2 y = 1 5) 2 x + 5 y = 10 10) 2 x – y = 3 Back to Main Page

What are the Co-ordinates of these points? 5 (x, y) 4 3 2 1 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 -2 -3 -4 -5 Back to Main Page

Negative Numbers Addition and Subtraction (1) 2 + 3 (2) 6 -5 (3) (5) -1 - 2 (6) -4 + 5 (7) (9) -3 + 6 (10) -4 - 1 3 -7 (4) -2 + 6 -2 - 2 (8) 0 – 4 (11) 6 - 8 (12) -5 - 2 (13) -8 + 4 (14) -5 - (- 2) (15) 0 - (- 1) (16) 7 - 12 + 9 (17) -4 - 9 + -2 (18) (19) -45 + 17 (20) 14 - (- 2) 4 - 5½ Back to Main Page

Negative Numbers Multiplication and Division (1) 4 x -3 (2) -7 x -2 (3) -5 x 4 (4) 28 ÷ -7 (5) -21 ÷ -3 (6) -20 ÷ 5 (7) -2 x 3 x 2 (8) -18 ÷ -3 x 2 (9) -2 x -2 (10) 2. 5 x -10 Back to Main Page

Substituting Numbers into Formulae Exercise Substitute x = 4 into the following formulae: 1) x– 2 2 6) 4 - 2 x -4 2) 2 x 8 7) -1 3) 3 x + 2 14 x-3 2 4) 1–x -3 8) 1 5) 3 – 2 x -5 3 -x 2 2 x – 6 9) Click forward to reveal answers 2 Back to Main Page

Substituting Negative Numbers into Formulae Exercise Substitute x = -1 into the following formulae: 1) x– 2 -3 6) 4 - 2 x 2) 2 x -2 7) 3) 3 x + 2 -1 x-3 2 4) 1–x 2 8) 5) 3 – 2 x 5 3 -x 2 2 x – 6 9) Click forward to reveal answers 6 -3½ 3½ -8 Back to Main Page