Calculate a gradient
Gradient is vertical change divided by horizontal change. Use two distinct points on the same straight line
\[\begin{gathered}m=\frac{y_2-y_1}{x_2-x_1}\\x_2\ne x_1\end{gathered}\]
- From a graph Read the scale on each axis. Choose two clear grid points and count rise and run in coordinate units
- From coordinates Subtract the vertical coordinates and the horizontal coordinates in the same point order. Divide the first difference by the second
- Equivalent choices Reversing both differences leaves the gradient unchanged. Choosing another pair on the same line also gives the same gradient
- Special cases Equal vertical coordinates give zero gradient when the points are distinct. Equal horizontal coordinates give undefined gradient
Worked example
Find the gradient of the line through P and Q.
\[\begin{aligned}P&=(-4,7)\\Q&=(2,-2)\end{aligned}\]
- \(\displaystyle \Delta y=-2-7=-9\)Subtract the vertical coordinates from P to Q. The negative change describes a fall
- \(\displaystyle \Delta x=2-(-4)=6\)Subtract the horizontal coordinates in the same order
- \(\displaystyle m=\frac{-9}{6}\)Divide vertical change by horizontal change
- \(\displaystyle m=-\frac{3}{2}\)Reduce the fraction by dividing numerator and denominator by their common factor
Common mistakes
Reversing the ratio
✕\(\displaystyle m=\frac{\Delta x}{\Delta y}\)✓\(\displaystyle m=\frac{\Delta y}{\Delta x}\)
Gradient uses vertical change per horizontal unit, so vertical change belongs in the numeratorReversing only one difference
✕\(\displaystyle m=\frac{y_2-y_1}{x_1-x_2}\)✓\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)
Both differences must describe the same movement between the pointsUsing one point alone
✕\(\displaystyle m=\frac{y_1}{x_1}\text{ for every line}\)✓\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)
A coordinate ratio works when the other point is the origin. In general, use the changes between two pointsDifficulty levels
introFind a positive integer gradient from a unit grid
standardFind a signed integer or half-integer gradient from a grid or a coordinate pair
challengeFind exact fractional gradients from labelled graphs or point pairs, including horizontal lines
examChoose an exact gradient from a graph or two coordinates
Questions about this topic
Must I start at the left point?+
That makes the run positive and is often simpler. Starting at the right point also works if both differences use that same order
Why can I use another pair on the same straight line?+
The vertical and horizontal changes scale by the same factor. Their quotient stays the same
Why is a vertical gradient undefined?+
Two points on a vertical line have zero horizontal change. The gradient calculation would divide by zero
Can the same point be used twice?+
No. One repeated point gives zero for both changes and does not determine a unique line
What if the graph has different numerical steps on its axes?+
Convert the counted intervals to coordinate units on each axis before dividing the vertical change by the horizontal change
Can I use a calculator?+
No. This topic is practised without a calculator
In this chapter
6A Reading linear graphs: Read and interpret a straight-line graphInterpret and sketch distance-time graphsConstruct a table of values from a formula6C Plotting lines from tables: Recognise a line or find points for its graph6C Plotting lines from tables: Read and interpret a straight-line graph6C Plotting lines from tables: Construct a linear model from a descriptionMidpoint and distance of a coordinate segment6E Gradient from coordinates: Classify and compare gradients from graphsFind the gradient of a line through two pointsCalculate a rate from a straight-line graphRead the vertical-axis intercept and gradient from an equation6G Gradient-intercept form: Recognise a line or find points for its graph6G Gradient-intercept form: Classify and compare gradients from graphsForm an equation of a parallel or perpendicular lineFind the axis intercepts of a line6I Sketching lines from intercepts: Recognise a line or find points for its graph6J Linear modelling: Construct a linear model from a descriptionInterpret the graph of a linear modelRead the point of intersection of two linesDirect proportion and the constant of proportionalityIdentify direct or inverse variation from tables and graphsFind the constant of variation and complete the table