0580 · 3.3Gradient of linear graphs

Gradient from a graph

0:00Two straight lines can rise across a graph at very different rates. One gains only a little height as you move right, while another climbs much faster. Gradient gives us a number that describes this difference. By the end, you will understand how to find it from a graph or from two coordinates. On a coordinate grid, x measures horizontal position and y measures vertical position. A point with x equal to two and y equal to five lies two units right and five units up from the origin.

0:26To calculate gradient, we compare changes in these positions. We need subtraction, division and careful handling of negative numbers. Gradient is the change in y divided by the corresponding change in x. It tells us how much y changes for each increase of one unit in x. Here, moving three units right takes us six units up. 6 ÷ 3 gives a gradient of two. A gradient of two means that y increases by two whenever x increases by one.

0:54The line does not need to pass through the origin for this to work. We measure changes between points, not their distances from the origin. The horizontal and vertical movements form what we call a gradient triangle. A straight line has the same gradient everywhere along it. Enlarging the gradient triangle multiplies both its horizontal and vertical sides by the same amount. Their ratio therefore stays unchanged. This is why any two distinct points on the same straight line give the same gradient.

1:20Our first exam-style question asks us to find the gradient of the line shown. Choose two clearly readable points that are well separated on the line. The first is the point one comma two, and the second is the point five comma eight. The change in y is 8 − 2, giving six. For the horizontal change, subtract one from five to get four. Dividing six by four gives 6/4.

1:43Simplifying gives 3/2, which is exactly one point five. Read the axis scales rather than simply counting squares. On this graph, each vertical interval represents two units, while each horizontal interval represents one. Three vertical intervals therefore represent six units, not three. A larger gradient triangle usually makes coordinate readings more reliable.

Gradient from coordinates

2:05When coordinates are supplied, the same idea becomes a formula. Gradient equals the second y coordinate minus the first, divided by the second x coordinate minus the first. The small numbers in the formula identify the two points. They do not mean that we multiply the coordinates by one or two. Our second example asks us to calculate the gradient through two given points. Point A is negative three comma five, and point B is one comma negative seven.

2:30Moving from A to B takes us rightwards and downwards, so we expect a negative gradient. The vertical change is −7 − 5, giving negative twelve. The horizontal change is 1 − −3. Subtracting negative three becomes adding three, so the horizontal change is four. −12 ÷ 4 gives a gradient of negative three. Read the direction of a line from left to right. A rising line has positive gradient, while a falling line has negative gradient.

3:00A horizontal line has zero gradient because its vertical change is zero. For a vertical line, the horizontal change is zero, and division by zero is undefined, so its gradient is undefined. The size of the gradient describes steepness when the axis scales are the same. A gradient of negative three is steeper than a gradient of negative one. The negative sign describes direction, not a lack of steepness. Avoid judging gradient from how high the line sits on the graph.

Consistent subtraction and exam tips

3:27For our final exam-style example, take P at negative two comma negative three, and Q at four comma one. The vertical change is 1 − −3, giving four. Horizontally, 4 − −2 gives six, so the gradient is 4/6, simplified to 2/3. Reversing the point order gives −4 ÷ −6, which still equals 2/3. Both changes reverse sign together, leaving the ratio unchanged. Mixing the point order between the numerator and denominator gives the wrong sign.

3:58Also, dividing horizontal change by vertical change answers a different question. These methods must work without a calculator on Paper two, and also apply on calculator Paper four. Show both coordinate differences and their division so your method is clear. Keep exact fractions when possible, and do not round intermediate values. If a decimal answer needs rounding, normally use three significant figures unless instructed otherwise. Gradient compares vertical change with horizontal change, not height with position.

4:27A straight line keeps this ratio constant wherever you choose your points. Read graph scales carefully, subtract coordinates in a consistent order, and retain the correct sign. That gives you both the gradient and an understanding of what it means.