Graphing linear relationships

Making a table of values from a rule, plotting real data with sensible axes, recognising a linear relationship, and reading values from the graph by interpolation and extrapolation.

Graphs in Standard maths are about real things, so the axes have names and units: dollars, litres, minutes, kilometres. A straight-line graph tells you the quantity changes at a steady rate. Once it's drawn, you can read answers straight off it.

From rule to table to graph

To graph a rule, choose a few $x$ values, work out each $y$, and plot the points. Three points are enough for a straight line: two to draw it and one to check it.

Example 1●

A $500$ L water tank is drained at a steady $25$ L per minute, so the volume left is $V = 500 - 25t$ after $t$ minutes. Make a table of values for $t = 0, 4, 8, 12, 16, 20$ and describe the graph.

$t$ (min)	$0$	$4$	$8$	$12$	$16$	$20$
$V$ (L)	$500$	$400$	$300$	$200$	$100$	$0$

Substitute each t. For t = 4: 500 − 100 = 400.

The graph is a straight line sloping down from $(0, 500)$ to $(20, 0)$.

Each step of 4 minutes drops V by the same 100 L, so the points lie on a straight line.

It only makes sense for $0 \le t \le 20$.

Time can't be negative and the volume can't go below 0, so only this part of the line makes sense.

The test for a linear relationship is in the table. If equal steps in $x$ always give equal steps in $y$, the points line up. If the steps in $y$ grow or shrink, the graph curves.

Plotting real data

Good graphs have a title, both axes labelled with units, an even scale on each axis, and points plotted with small crosses or dots. The input variable (the one you control, often time) goes across. The output goes up.

InteractivePlot it, then read it

Try this

Choose the tank. Tap the grid to plot each point from the table. When all six are right, the line appears. Drag the reader to find when $250$ L is left. Then pick the savings plan, plot it, and drag the reader past the last point. Why does the tool warn you there?

Reading from a graph

Interpolation means reading a value between your data points. On a straight-line model this is usually reliable.
Extrapolation means extending the line beyond your data. It assumes the pattern keeps going, which often isn't true.

Example 2●●

Use the tank graph from Example 1. (a) How much water is left after $10$ minutes? (b) When will $150$ L be left?

(a) $V = 250$ L. Check: $500 - 25 \times 10 = 250$.

Go across to t = 10 and up to the line. It's halfway between 300 and 200.

(b) $t = 14$ minutes. Check: $500 - 25 \times 14 = 150$.

Go up the V axis to 150, across to the line, then down.

Example 3●●

A baby's mass was $3.4$ kg at birth and $6.2$ kg at $4$ months, and it grew close to linearly in between. A parent extends the line to predict the mass at $10$ years. Comment.

Rate of growth $= \dfrac{6.2 - 3.4}{4} = 0.7$ kg per month.

The rate is 2.8 kg over 4 months, or 0.7 kg per month.

Extrapolating: $3.4 + 0.7 \times 120 = 87.4$ kg.

Ten years is 120 months.

A $10$-year-old weighs about $30$ kg, not $87$ kg. Extrapolating far beyond the data gave a silly answer.

Babies' growth slows down a lot after the first year, so the model fails outside the data.

Your turn

A candle is $24$ cm tall and burns down $1.5$ cm each hour, so $h = 24 - 1.5t$. Complete the table for $t = 0, 4, 8, 12, 16$ and find when the candle burns out.

Show the working

$h$: $24, 18, 12, 6, 0$. Each $4$ hours removes $6$ cm. It burns out when $h = 0$, at $t = 16$ hours.

Flashcards

How to spot a linear table

Equal steps in $x$ give equal steps in $y$

Interpolation

Reading between the data points: usually reliable

Extrapolation

Extending beyond the data: assumes the pattern continues, so be careful

Practice

For $y = 3x - 3$, find $y$ when $x = 4$.
Which table shows a linear relationship?
Using $V = 500 - 25t$, how many litres are left after $10$ minutes?
Using $V = 500 - 25t$, after how many minutes are $150$ L left?
Estimating a value between two plotted data points is called:
On a graph of cost against time, which variable usually goes on the horizontal axis?
A candle's height is $h = 24 - 1.5t$ cm after $t$ hours. After how many hours does it burn out?
A tradie's charges lie on a straight line: $\$90$ for $0$ hours and $\$390$ for $4$ hours. What is the charge, in dollars, for $2$ hours?
Exam style: A plant grew from $5$ cm to $17$ cm in $6$ weeks at a steady rate. Using a linear model, which prediction is least reliable?
Exam style: A plant grew from $5$ cm to $17$ cm in $6$ weeks at a steady rate. Use a linear model to estimate its height, in cm, at $3$ weeks.