Gradient of a line: worked examples and method

Rise over run

The gradient of the line through (1, 2) and (5, 10) is 2.

Sign

A line through (1, 1) and (3, 5) has a positive gradient.

Change in y

The change in y from (1, 2) to (5, 10) is 8, before you divide.

From the axis

A line has gradient 3 and passes through (0, 1), so y = 3x + 1. When x=2, y = 3 × 2 + 1 = 7.

Order

Subtract the first point from the second in both x and y.

Not the intercept

The gradient is the multiplier of x, not the crossing on the y-axis.

From two coordinates

Pick an order and stick to it. From (1, 2) to (5, 10), the change in y is 10-2=8 and the change in x is 5-1=4. The gradient is 8/4=2. Subtracting y in one order and x in the other flips the sign: (2-10)/(5-1)=-2, which is wrong, because this line rises from left to right. Use the same "second minus first" on the top and the bottom.

Simplify the fraction. 8/4=2. An integer or a cancelled fraction is the usual finished answer. A decimal such as 0.333 for 1/3 loses accuracy.

Method: 1. Pick one direction along the line and use that same direction for both changes. 2. From (1, 2) to (5, 10), the change in y is 8 and the change in x is 4. 3. Divide those changes: the gradient is 8/4=2.

From a drawn line

Draw a right-angled triangle under a section of the line, using grid lines for the two legs. Make the triangle large. A one-square triangle magnifies a reading error. Label the vertical length and the horizontal length in the units of the axes, not in "number of squares", unless each square is worth 1. If one square is 2 units across and 5 units up, those scale factors go into the lengths before you divide.

Look at the direction before you attach a sign. If the line falls from left to right, the gradient is negative even when both lengths you measured are positive distances. The triangle gives you sizes. The direction gives you the sign. A horizontal line gives a rise of 0, so the gradient is 0. A vertical line has no run, and its gradient is undefined. You will not be asked for a finite gradient of a vertical line.

Method: 1. Draw a right-angled triangle on the line, with both legs running along the grid. 2. Make the triangle several squares wide so the reading is stable. 3. Avoid a triangle that is only one square wide, because a small slip changes the gradient a lot.

What the number means

On a distance-time graph with time in hours and distance in kilometres, a gradient of 4.5 is 4.5 km/h. On a cost graph it is the price per item, after you have accounted for any standing charge by using two points rather than one point and the origin. Parallel lines, such as y=2x+1 and y=2x-4, have the same gradient, here 2. You can say they are parallel without drawing them.

A negative gradient in a story is a decrease: a falling temperature, a draining tank, a journey back towards the start if distance is measured from home. State the unit. A bare -2 might be correct arithmetic and still miss the mark if the question asked for a speed or a rate.

Method: 1. On a distance-time graph measured in kilometres and hours, a gradient of 4.5 means 4.5 km/h. 2. On a cost graph, use two points on the sloping line so a standing charge is not mistaken for the rate. 3. Parallel lines such as y=2x+1 and y=2x-4 share one gradient, here 2.

Reading the gradient from an equation

When a line is written as y = mx + c, the gradient is the number m in front of x. So y = 4x - 1 has gradient 4 and y = 7 - 3x has gradient -3. Take care when the x term is written second: the sign belongs to the number, so the gradient of y = 7 - 3x is negative three, not seven. Two lines with the same gradient are parallel.

If the equation is not in that form, rearrange it. The line 2y = 10x + 6 becomes y = 5x + 3 after dividing every term by 2, so the gradient is 5. For 3x + y = 7 rearrange to y = 7 - 3x to get a gradient of -3. Dividing only the x term is a common slip, so divide every term. Check by finding two points on the line and calculating the rise over the run.

Method: 1. Rearrange the equation so that y is on its own on the left. 2. Divide every term by the number in front of y. 3. Read the gradient as the coefficient of x, including its sign. 4. Compare gradients to decide whether two lines are parallel.

Tip: The gradient of y = 7 - 3x is -3. The sign stays with the number it is attached to.

Perpendicular lines and right angles

Two lines are perpendicular when their gradients multiply to give -1. Equivalently, the perpendicular gradient is the negative reciprocal. A line with gradient 2 is perpendicular to a line with gradient -1/2, because 2 × -1/2 = -1. To find a negative reciprocal, turn the fraction upside down and change its sign.

A frequent exam question gives three points and asks you to show that the angle at one of them is 90°. Find the gradient of each of the two sides that meet at that point. If the product of the gradients is -1, the sides are perpendicular and the angle is a right angle. Show both gradients, show the product, and finish with a sentence that states the conclusion. If the product is not -1, the angle is not a right angle.

Method: 1. Find the gradient of each of the two lines that meet at the angle. 2. Multiply the two gradients together. 3. If the product is -1, the lines are perpendicular. 4. Write the conclusion in words, naming the angle.

What the gradient means in a story

On a graph with real units, the gradient is the change in the quantity on the vertical axis for each one unit on the horizontal axis. On a distance-time graph with kilometres and hours it is a speed in km per hour. On a cost graph it is the price for each extra item or hour. The command is often "Describe fully what the gradient represents", so name the quantity and the units.

A good answer is a short sentence such as "the speed of the cyclist is 10 km per hour" or "the cost of each extra hour is £5". If the line does not start at zero, the other number matters too. The place where the line meets the vertical axis is the starting value, such as a fixed charge. A steeper line has a larger gradient, so it shows a faster speed or a higher price rate.

Method: 1. Pick two points on the line and write down their coordinates with units. 2. Divide the change in y by the change in x. 3. Write the units as vertical units per horizontal unit. 4. Say in a sentence what the number represents in the story.

Tip: Include the units in the gradient, for example 10 km per hour. A bare number is not a full description.

Finding a missing coordinate from a gradient

If a question tells you the gradient and gives one point and part of another, set up an equation. For a line through (1, 3) and (5, k) with gradient 2, write (k - 3)/(5 - 1) = 2. Multiply both sides by 4 to get k - 3 = 8, so k = 11. Always keep the same order of subtraction on the top and the bottom.

You can also use a gradient to show that three points lie on one straight line. Find the gradient from the first point to the second and from the second to the third. If the two gradients are equal, the points are on a straight line. A vertical line has no gradient, because the run is zero and you cannot divide by zero. A horizontal line has gradient zero.

Method: 1. Write the gradient formula with the unknown in it. 2. Multiply both sides by the run to remove the fraction. 3. Solve the equation for the unknown. 4. Check by working out the gradient with your answer.