Calculate a gradient
Gradient is vertical change divided by horizontal change. Use two distinct points on the same straight line
\[m=\frac{y_2-y_1}{x_2-x_1},\qquad x_2\ne x_1\]
- 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
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
✕\(\displaystyle m=\frac{y_1}{x_1}\)✓\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)
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 I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 4: Linear relations
In this chapter
Construct a table of values from a formulaRead coordinates and points from a straight-line graph4A Points, tables and linear relations: Decide whether a point lies on a lineFind the axis intercepts of a line4C Horizontal, vertical and through-origin lines: Recognise horizontal, vertical and through-origin graphsClassify and compare gradients from graphsFind the gradient of a line through two points4E Direct proportion: Use a direct-proportion model4E Direct proportion: Read and interpret direct-proportion graphsRead the vertical-axis intercept and gradient from an equation4F Gradient-intercept form: Recognise a line or find points for its graphConstruct a line on a coordinate gridFind the equation of a line from its graph or from its intercept and slopeRecover an intercept or recognise a line from a gradient and pointFind the equation of a line through two pointsMidpoint, endpoint and distance of a coordinate segmentForm an equation of a parallel or perpendicular lineDetermine a perpendicular bisector (optional extension)4J Linear modelling: Construct a linear model from a descriptionInterpret the graph of a linear model4K Graphical simultaneous equations: Check a candidate point against two lines (optional extension)Read the point of intersection of two lines (optional extension)