Gradients and y = mx + c

The gradient of a straight line tells you how steep it is. The equation y=mx+c tells you the gradient of a line and where it crosses the y-axis, so you can read facts about a line straight from its equation.

In the exam you are usually asked to write down the gradient of a line such as y=2x+5, or the point where it crosses the y-axis, for 1 mark each. Questions worth 2 or 3 marks ask you to work out a gradient from a graph or from two points, or to find an equation for a line drawn on a grid.

Other questions ask about parallel lines, or give you the gradient and ask for a missing coordinate. These come near the end of a paper, and the examiners say some students don't attempt them at all. The steps on this page can win you those marks.

The gradient is how far a line goes up for every 1 it goes across. A line with gradient 3 goes up 3 for every 1 across. A line with gradient \frac{1}{2} goes up 1 for every 2 across, so it is less steep.

To work out a gradient, pick two points on the line and use

\text{gradient}=\frac{\text{change in }y}{\text{change in }x}

The change in y is how far up or down you go. The change in x is how far across you go, from left to right.

A line that goes up from left to right has a positive gradient. A line that goes down from left to right has a negative gradient. A flat line has a gradient of 0, and its equation is y= a number, such as y=-2

On a graph, choose two points where the line goes exactly through a corner of a grid square. Draw a right-angled triangle from one point to the other, then count the squares across and up. Check the scale on each axis first: one square does not always mean 1.

You don't need a graph when you know two points on the line. Subtract the y-coordinates to find the change in y. Subtract the x-coordinates, in the same order, to find the change in x. Then divide.

For (1,3) and (4,9): the change in y is 9-3=6 and the change in x is 4-1=3, so the gradient is 6\div3=2

Keep the same order on the top and the bottom. If the change in y is negative, the gradient is negative and the line goes down. Put a negative coordinate in a bracket: 3-(-1)=3+1=4

The rule also works backwards, to find a missing coordinate. If you know the gradient and the change in x, then

\text{change in }y=\text{gradient}\times\text{change in }x

A gradient of 2 with a change in x of 5 gives a change in y of 2\times5=10. If the missing number is an x-coordinate, divide instead: \text{change in }x=\text{change in }y\div\text{gradient}

The equation of a straight line can usually be written as

y=mx+c

where m is the gradient and c is the intercept: the value of y where the line crosses the y-axis.

For y=4x+1, the gradient is 4 and the line crosses the y-axis at (0,1). A point on the y-axis always has x-coordinate 0, so the line crosses at (0,c)

Move the c slider and the whole line slides up or down, keeping the same steepness, with P always at (0,c). Move the m slider and the line turns about P. A bigger positive m makes it steeper, a negative m makes it go down from left to right, and m=0 makes it flat.

Under the graph, the rise from P to Q is divided by the run of 2. Whatever you choose, this gradient always comes out as m. The graph writes 1.5 as the fraction \frac{3}{2}, which is the same number.

The x term doesn't have to come first. In y=5-2x the number in front of x is -2, so the gradient is -2 and the intercept is 5. Take the sign in front of the x term with it.

Parallel lines go in the same direction and never meet. Parallel lines have the same gradient. So y=3x+1, y=3x-4 and y=3x are all parallel: each has gradient 3, with a different intercept.

To read m and c, the equation must start y=, with y on its own. If it starts 2y=, divide every term by 2 first. Dividing 2y=6x+4 by 2 gives y=3x+2, so the gradient is 3, not 6.

If the x term is on the same side as y, move it first. Subtracting 2x from both sides of y+2x=7 gives y=7-2x, so the gradient is -2.

• Intercept: find where the line crosses the y-axis. That value of y is c.
• Gradient: pick two points on the line and work out \text{change in }y\div\text{change in }x. That is m.
• Equation: put both numbers into y=mx+c.

Your answer must be an equation that starts y=, such as y=2x-3. Coordinates such as (0,-3) are not an equation, so they score nothing. Writing the line's name in place of y, such as L=2x-3, loses the final mark.

• Giving coordinates instead of an equation. An equation of a line starts y=, such as y=-2x+3.
• Missing the minus sign. A line that goes down from left to right has a negative gradient.
• Reading the wrong number from the equation. In y=3-4x the gradient is -4, the number in front of x.
• Not getting y on its own first. For 2y=6x+4, divide every term by 2, so the gradient is 3.
• Stopping after drawing the triangle. Count the change in y and the change in x, then divide.
• Counting squares without checking the scale. If one square is 2 on an axis, double the number of squares.