Classify and compare gradients
Read direction from left to right. Use equal horizontal distances to compare steepness
- Positive and negative Rising means positive. Falling means negative. A line's position above or below an axis does not decide its gradient
- Zero and undefined Horizontal lines have zero gradient. Vertical lines have undefined gradient
- Steepness Compare the sizes of the vertical changes. A larger fall can be steeper than a smaller rise
Worked example
Which line is steeper? The given changes for J and K are vertical
\[\begin{aligned}\text{Both runs}&=3\\\text{Change for J}&=-8\\\text{Change for K}&=5\end{aligned}\]
- \(\displaystyle |-8|=8\)Use the size of the fall for line J
- \(\displaystyle 8>5\)The horizontal changes match. Line J has the larger vertical change in size, so it is steeper
Common mistakes
Using height as gradient
✕\(\displaystyle \begin{gathered}\text{below the axis}\\\Rightarrow\text{negative gradient}\end{gathered}\)✓\(\displaystyle \begin{gathered}\text{falls from left to right}\\\Rightarrow\text{negative gradient}\end{gathered}\)
Height is a position. Gradient describes a change as you move rightSwapping the special cases
✕\(\displaystyle \text{vertical gradient}=0\)✓\(\displaystyle \text{vertical gradient is undefined}\)
A vertical line has zero horizontal change. A horizontal line has zero vertical changeIgnoring the size of a fall
✕\(\displaystyle \begin{gathered}-4<1\\\Rightarrow\text{the rising line}\\\text{is steeper}\end{gathered}\)✓\(\displaystyle |-4|>|1|\)
Compare gradient sizes when deciding steepness. A falling line can be steeperDifficulty levels
introClassify rising and falling lines
standardClassify all four types of gradient and compare clearly different steepnesses
challengeCompare closer fractional gradients, including rising and falling lines through one crossing
examChoose a classification or a steepness comparison from a labelled graph
Questions about this topic
Can a falling line be steeper than a rising line?+
Yes. Compare the sizes of their vertical changes over the same horizontal distance
Does moving a line upwards change its gradient?+
No. Moving the whole line upwards keeps its direction and steepness
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)