Search topics

Search

Read and interpret a straight-line graph

Start a set of 8
Step 7 of 18
Candle model through measured endpoints (0 hours,20 cm) and (4 hours,10 cm). The solid known interval is 0 to 4 hours; the axis extends to 8 hours.246848121620240Time (h)Height (cm)

Assume the candle continues losing 2.5 cm each hour from its initial 20 cm. Predict its height at 6 h in centimetres

Year 10, Chapter 6, 4 difficulty levels, No calculatorBuild a worksheet

Read a straight-line graph

A coordinate is an ordered pair: horizontal input first, vertical output second. A graph can support forward readings, reverse readings and signed changes. A model prediction is limited by its known interval and physical meaning

\[y=mx+c\]
  1. Identify and convert Read both axis titles and scales. Convert a physical quantity to graph units before substituting it as y
  2. Read forwards or backwards From x, trace vertically to the line then horizontally to y. From y, reverse this route
  3. Check the interval Between known observations is interpolation. Beyond them is extrapolation, even if the input is still visible within the graph window
  4. Calculate a change Subtract the earlier value from the later value. A difference between percentages is measured in percentage points

Worked example

A line has graph rule y=2x+1. Each vertical graph unit represents 10 L, and the known time interval is 1 to 3 minutes. When would the model give 90 L?

\[y=2x+1\]
  1. \(\displaystyle y=90/10=9\)Convert litres to graph units
  2. \(\displaystyle x=(9-1)/2=4\)Read backwards by rearranging the rule; the model gives 90 L at 4 minutes
  3. \(\displaystyle 4>3\)This is extrapolation beyond 3 minutes. It assumes the linear model continues and remains physically possible

Common mistakes

Ignoring the scale
✕\(\displaystyle 2(3)+1=7\text{ L}\)✓\(\displaystyle 10(2(3)+1)=70\text{ L}\)
The coordinate 7 represents 70 L on this scaled graph
Replacing a change by an endpoint
✕\(\displaystyle \Delta C=55\)✓\(\displaystyle \Delta C=55-85=-30\)
The change is −30 percentage points, not the final charge 55%
Treating a formula as unlimited evidence
✕\(\displaystyle h(10)=-5\text{ cm}\)✓\(\displaystyle 0\le t\le8\)
For h=20−2.5t, the physical candle model ends at 8 h when its height reaches zero

Difficulty levels

introRead values and signed changes, then classify a separate input
standardRead an input from a stated output and classify the known interval
challengeReverse or extend a line, including vertical scale conversion
examSelect an exact value or signed change from distinct alternatives

Questions about this topic

Are x and y always time and height?+

No. x is conventionally horizontal and y vertical, but their meanings and units depend on the axis titles

Is an endpoint interpolation?+

An endpoint is a known boundary value. Strict interpolation concerns values between known observations; extrapolation goes beyond them

Does a wider graph window change interpolation into extrapolation?+

No. Use the stated known interval. Enlarging the window does not create observations. Practice here explicitly treats its drawn segment as the known interval

How do I reverse a scaled graph reading?+

Convert the stated output into graph units first, then read horizontally to the line and down to the input axis

Is a falling line a negative quantity?+

Not necessarily. A positive quantity can decrease. Its signed change is negative even while both endpoint values remain positive

Can I always keep extending a straight line?+

No. Check whether the same process continues and whether the result respects bounds such as zero, 100% or capacity

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 6: Linear graphs

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