Horizontal, vertical and through-origin graphs
Read which coordinate is fixed, or use the origin and a second point. These descriptions can overlap at the axes
- Horizontal The vertical coordinate stays fixed and the gradient is zero \(\displaystyle y=c\)
- Vertical The horizontal coordinate stays fixed and the gradient is undefined \(\displaystyle x=k\)
- Through the origin A zero constant puts the origin on the graph. Check another distinct point \(\displaystyle y=mx\)
- Axis cases The horizontal axis is horizontal and passes through the origin. The vertical axis also passes through the origin but has no finite gradient \(\displaystyle \begin{aligned}y&=0\\x&=0\end{aligned}\)
Worked example
Find two distinct points to check the graph
\[y=-\frac{3}{2}x\]
- \(\displaystyle y(0)=-\frac{3}{2}(0)=0\)The origin is one point
- \(\displaystyle y(2)=-\frac{3}{2}(2)\)Choose a nonzero horizontal coordinate that keeps the arithmetic simple
- \(\displaystyle y(2)=-3\)Calculate the vertical coordinate
- \(\displaystyle (0,0),\quad(2,-3)\)Check that the candidate contains both points and falls from left to right
Common mistakes
Fixing the wrong coordinate
✕\(\displaystyle x=4\text{ is horizontal}\)✓\(\displaystyle x=4\text{ is vertical}\)
The horizontal coordinate is fixed while the vertical coordinate variesUsing the origin twice
✕\(\displaystyle \begin{gathered}(0,0),(0,0)\\\text{two distinct points}\end{gathered}\)✓\(\displaystyle \begin{gathered}(0,0),(2,-3)\\\text{two distinct points}\end{gathered}\)
For the worked rule, the two axis intercepts coincide. Use a different point to determine the directionGiving a vertical line zero gradient
✕\(\displaystyle \text{vertical gradient}=0\)✓\(\displaystyle \text{vertical gradient is undefined}\)
Two different points on a vertical line have zero horizontal changeDifficulty levels
standardMatch horizontal, vertical and through-origin equations to graphs
examRecognise the same line forms in a short graph-choice question
Questions about this topic
Can a line belong to more than one special case?+
Yes. The horizontal axis is both horizontal and through the origin
Does every line have a vertical-axis intercept?+
A vertical line away from the vertical axis has none. The vertical axis itself has no unique intersection with that axis
How do I sketch a special line on paper?+
For a fixed-coordinate line, mark two points with that coordinate and join them with a ruler. For an origin line, use the origin and another distinct point satisfying the rule. Extend the line in both directions
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)