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This page sets out the method for drawing linear graphs in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Drawing linear graphs topic page.
When you see 'draw the graph of … for values of from … to …', make a table.
When the equation is in the form , find the two axis crossings quickly.
Pick three x-values, often −1, 0 and 1, or 0, 1 and 2.
Work out y for each one.
Plot the points and join them with a ruler.
If they are not in a straight line, one y-value is wrong.
m is how much y increases when x increases by 1.
A gradient of 2 goes up 2 for every 1 across.
c is the y-value when x = 0.
You can plot that point before any other.
A negative gradient slopes down as x increases.
y = −x + 4 crosses the y-axis at 4 and falls.
A point on the line satisfies the equation.
If x = 3 on y = 2x + 1, then y = 7.
Choose the x-values the question lists, or three easy ones such as -1, 0 and 2 if you are choosing. Substitute each one separately. For y=3x-2 at x=-1, write 3×(-1)-2=-3-2=-5. At x=0, y=-2. At x=2, y=4. Zero is worth plotting because it gives the y-intercept with no multiplication at all.
Look along the y row before you plot. For a straight line the y-values go up or down by the same amount each time x increases by 1. In y=3x-2 that step is +3. If one value breaks the pattern, recalculate that column only. Plotting a broken pattern produces a kink, and a kink means the graph is no longer the line you were asked for.
Method: 1. Use the x-values the question lists, or choose -1, 0 and 2 when you may pick. 2. Substitute one x-value at a time. 3. For y=3x-2 at x=-1, calculate 3×(-1)-2=-5.
Plot (x, y) by going along first, then up or down. Use a small cross, not a blob, so the point sits on the intersection of the grid lines. Then place a ruler along the crosses and draw one straight line. Extend it across the whole range of x the question names. Stopping at the first and last plotted points leaves the line short, and a short line cannot be read at the edge of the grid.
A wobbly freehand join is not a straight line, and it loses the accuracy mark even when the table was right. If the question puts two lines on the same grid, label each line with its equation. A reading taken from the wrong line is a different pair of coordinates.
Method: 1. Plot each pair by going along the grid first, then up or down. 2. Mark a small cross on the grid lines so the point is exact. 3. Place a ruler along the crosses and draw one straight line.
An equation such as 2x+y=10 can be rearranged before the table. Subtract 2x from both sides: y=10-2x. Now it is in the form you already plot. At x=0, y=10. At x=1, y=8. At x=3, y=4. At x=5, y=0. The step in y is -2 each time x increases by 1, which matches the -2x.
The cover-up method finds the two axis crossings without a full table. Cover y in 2x+y=10 and solve 2x=10, so the line crosses the x-axis at (5, 0). Cover x and solve y=10, so it crosses the y-axis at (0, 10). Two points decide a straight line. A third point is still worth calculating, because it checks the first two. Reading the line afterwards is ordinary coordinate reading: go from the axis to the line, then to the other axis.
Method: 1. Rearrange an equation such as 2x+y=10 before you build the table. 2. Subtract 2x from both sides to reach y=10-2x. 3. Plot that rearranged form in the same way as any other y=ldots equation.
A real exam question usually gives a partly filled table and asks you to complete it before you draw the line. The work is the same every time: put each x into the equation in brackets and work out y. Negative values of x are where marks go. For y=2x-3 and x=-1 the calculation is 2×(-1)-3=-2-3=-5, not -1.
Equations written as y=4-x or y=6-2x need extra care. Subtracting a negative adds, so y=6-2x with x=-2 gives 6-2×(-2)=6+4=10. Look for patterns as a check. For a straight line the y values go up or down by the same amount each time x goes up by one. If your row jumps by 2, then 2, then 5, one value is wrong.
Method: 1. Write the equation and the x value in brackets, such as 6-2×(-2). 2. Work out the multiplication first and then the addition or subtraction. 3. Fill in every blank in the table and check that y changes by a constant step. 4. Write each pair as coordinates (x, y) before you plot.
Tip: In a table for a straight line, the step in y equals the number in front of x in the equation.
Plot each point as a small cross using a sharp pencil, going across to the x value first and then up or down to the y value. Check the scale on each axis before you start, because the squares are not always worth 1. A straight line graph must be drawn with a ruler. The points should lie exactly on one line, so if one point is out of line, go back and check the table.
Extend the line across the whole range given in the question, for example "for values of x from -1 to 3". Do not stop at your first and last crosses if the axes go further and the question does not limit the range. Mark schemes usually give two marks for the correct line and one mark for at least two correct points, so a neat line with one slip still scores.
Method: 1. Check the scale on both axes. 2. Plot every point from the table with a small, neat cross. 3. Use a ruler to join the points and extend the line across the whole range. 4. Check that the line passes through a point you did not use, such as when x=0.
A line in the form y=mx+c has gradient m and crosses the y-axis at (0, c). For y=5x-4 the gradient is 5 and the line crosses the y-axis at (0, -4). Notice that the coordinates need both numbers, not just -4. When the line is written as y=4-3x the gradient is -3 because the term in x has a minus sign, and the intercept is (0, 4).
Lines with the same gradient are parallel, so y=2x-1 is parallel to y=2x+5. To find where a line meets the x-axis, put y=0 into the equation and solve. For y=2x+1 this gives 0=2x+1, so x=-0.5. Questions may ask for either intercept as coordinates, and the answer should be written in a bracket with the zero included.
Method: 1. Rearrange the equation to the form y=mx+c if it is not already. 2. Read off the gradient m and the intercept c, including any minus signs. 3. Write the y-intercept as the pair (0, c). 4. For an x-intercept put y=0, solve for x and write (x, 0).
Tip: Parallel lines have the same m and different c.
Equations such as x+2y=10 or 2y=10x are not yet in the form y=mx+c. Rearrange them by doing the same to both sides. From 2y=10x divide by 2 to get y=5x. From x+2y=10 subtract x and divide by 2 to get y=-1/2x+5. A quick way to find the intercepts is to cover up one letter: put x=0 for the y-intercept and y=0 for the x-intercept.
To test whether a point is on a line, substitute its coordinates. For y=2x+1 and the point (4, 10) the right side is 2×4+1=9, which is not 10, so the point is not on the line. Always write the value you get and compare it with the given y value. A correct conclusion with no substitution shown earns no marks.
Method: 1. Isolate y by doing the same operation to both sides. 2. Write the new equation clearly, then read off the gradient and intercept. 3. To test a point, replace x with its value and work out y. 4. State whether the calculated y matches, and write a conclusion.
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