Straight-line graphs

An equation like y = 2x − 1 gives a straight line: a linear graph.

1. Make a table of values for the x-values you're given.
2. Work out y for each x.
3. Plot the points and join them with a ruler, right to the ends.

Example: draw y = 2x − 1 for −2 ≤ x ≤ 3.

1. x = −2: y = 2 × −2 − 1 = −5
2. Work out each y the same way, then plot the points.
3. They lie in a straight line

If one point doesn't line up with the others, recheck its y-value.

The gradient is how steep a line is: how far it goes up for every 1 across.

Gradient = change in y ÷ change in x (rise over run). Learn this.

A positive gradient slopes up from left to right; a negative one slopes down.

Example: find the gradient of the line through (1, 3) and (4, 12).

1. Change in y: 12 − 3 = 9
2. Change in x: 4 − 1 = 3
3. Gradient = 9 ÷ 3 = 3

Example: find the gradient of the line through (−2, 5) and (2, −3).

1. Change in y: −3 − 5 = −8
2. Change in x: 2 − (−2) = 4
3. Gradient = −8 ÷ 4 = −2

Every straight line (except a vertical one) has an equation y = mx + c:

• m is the gradient
• c is the y-intercept, where the line crosses the y-axis.

To read m and c, the equation must start "y =".

Example: give the gradient and y-intercept of y = 3x − 2.

1. m = 3 and c = −2
2. Gradient 3, crossing the y-axis at (0, −2)

Example: find the gradient and y-intercept of 3x + 4y = 12.

1. Subtract 3x: 4y = −3x + 12
2. Divide by 4: y = −¾x + 3
3. Gradient −¾, y-intercept 3

To find y = mx + c, you need m, then a point on the line to work out c.

1. Find the gradient m.
2. Substitute m and one point into y = mx + c, and solve for c.

Example: find the equation of the line with gradient 4 through (2, 5).

1. 5 = 4 × 2 + c, so 5 = 8 + c
2. c = −3, so y = 4x − 3

Example: find the equation of the line through (2, 1) and (5, 10).

1. m = (10 − 1) ÷ (5 − 2) = 9 ÷ 3 = 3
2. Using (2, 1): 1 = 3 × 2 + c, so c = −5
3. y = 3x − 5

Check with the other point: 3 × 5 − 5 = 10.

• A horizontal line has the same y-value everywhere: its equation is y = b. Its gradient is 0.
• A vertical line has the same x-value everywhere: its equation is x = a.

The x-axis is y = 0, and the y-axis is x = 0.

Example: find the equation of the line through (4, −1) and (4, 6).

1. Both points have x = 4.
2. The line is vertical.
3. x = 4