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Read and interpret direct-proportion graphs

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A straight segment on volume-versus-time axes goes from zero minutes and zero litres to four minutes and sixteen litres. A point is marked at two minutes and eight litres12343691215180Time t (min)Volume V (L)

Which rule matches the graph?

Year 9, Chapter 4, 4 difficulty levels, Calculator allowedBuild a worksheet

Direct-proportion graphs

Use the origin, a distinct point and the labelled scales to connect the graph to a constant rate

\[y=kx\]
  1. Rate equals gradient The vertical amount divided by the corresponding nonzero horizontal amount gives the rate for an origin line \(\displaystyle k=\frac{y}{x}\quad(x\ne0)\)
  2. Keep the units A time axis in minutes and a volume axis in litres give a rate in litres per minute. Convert a requested quantity to the axis unit before reading the graph \(\displaystyle 120\text{ seconds}=2\text{ minutes}\)
  3. Construct on paper Label both axes and their units, choose consistent scales, plot two distinct points, and join them with a straight edge over the supported interval
  4. Check both conditions A line away from the origin is not direct proportion. A curved graph through the origin is not direct proportion either

Worked example

A delivery-volume graph is straight from the origin to the stated endpoint. Find the rate and write the rule. Then identify two points for drawing the modelled interval on paper

\[\begin{gathered}\text{endpoint}=(5,35)\\t\text{ in minutes},\quad V\text{ in litres}\\0\le t\le5\end{gathered}\]
  1. \(\displaystyle k=\frac{35-0}{5-0}\)Divide vertical change by horizontal change
  2. \(\displaystyle k=\frac{35}{5}\)Evaluate the differences while preserving their order
  3. \(\displaystyle k=7\text{ litres per minute}\)State the numerical rate with the axis units
  4. \(\displaystyle V=7t\)The origin supplies the zero starting value
  5. \(\displaystyle (0,0),\quad(5,35)\)These are distinct points. Plot them on labelled scales and join only across the stated interval

Common mistakes

Reversing the rate
✕\(\displaystyle k=\frac{5}{35}\)✓\(\displaystyle k=\frac{35}{5}\)
In the worked model, volume is vertical and time is horizontal
Using the origin twice
✕\(\displaystyle \begin{gathered}(0,0),(0,0)\\\text{determine one line}\end{gathered}\)✓\(\displaystyle \begin{gathered}(0,0),(5,35)\\\text{determine one line}\end{gathered}\)
For the nonzero rate in the worked example, the two intercepts coincide, so a separate point is needed
Testing only the origin
✕\(\displaystyle y=x^2\text{ is direct proportion}\)✓\(\displaystyle y=7x\text{ is direct proportion}\)
Both pass through the origin, but only the straight-line rule keeps a constant ratio

Difficulty levels

introRead an output from a direct-proportion graph
standardRead an input or calculate the rate with the axis units
challengeRead an output and convert it to the requested unit
examSelect a direct-proportion graph from its rule

Questions about this topic

Can I compare slopes by appearance alone?+

Only when both graphs use the same scales and aspect ratio. Reading coordinate changes works even when the axes have different unit scales

Does selecting a graph show that I can draw it?+

Selection checks recognition. Drawing also requires choosing scales, plotting accurately and showing the relevant interval

Can I use a calculator?+

A calculator is allowed unless a question is marked no 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)