4 Graphs

Learning objectives

44A Graphical and numerical formsTranslate information between graphical and numerical form.44C Plot two variablesPlot two variables, discrete or continuous, from experimental or other data.44D Slope and interceptDetermine the slope and intercept of a linear graph.

Translating between graphical and numerical form preserves the variables, units and scale while changing how the relationship is represented. Read coordinates from the axes before describing a pattern.

Graph-to-number move	Number-to-graph move
identify the horizontal variable, vertical variable and units	put each paired observation into an $(x,y)$ row
locate a stated $x$ value and move vertically to the plotted point or line	choose an axis range and regular scale covering all values
move horizontally to read the corresponding $y$ value	plot each pair at its coordinate
calculate a difference or rate from selected coordinates when required	use a key when several datasets share the axes
describe direction, range, plateau or fluctuation with values	retain the exact table values even when the graph summarises their pattern

If a myelinated neurone graph gives 4.4 m s⁻¹ at a diameter of 1.0 µm, the numerical translation is the coordinate $(1.0\,\mu\mathrm{m},4.4\,\mathrm{m\,s^{-1}})$ . State both values and units.

Reading between plotted values is interpolation; reading beyond the measured range is extrapolation and is less secure. Do not infer a value from visual height without checking the scale increments.

A two-variable graph places the independent variable on the horizontal axis and the dependent variable on the vertical axis so every paired observation has one unambiguous coordinate.

Plotting decision	Required feature
choose axes	independent variable on $x$ ; dependent variable on $y$
label completely	variable name and unit on each axis
set scales	linear, regular increments that use much of the available grid and include all values
plot points	small, accurate marks at every coordinate
represent connection	join ordered continuous measurements only when instructed or scientifically justified; otherwise use a suitable best-fit line or leave points unjoined
compare datasets	use distinguishable lines or symbols and a complete key; do not extrapolate beyond the data

For seedling dry mass measured on days 4, 8, 12, 16 and 20, plot day on $x$ and dry mass in g on $y$ . Plot fertiliser and no-fertiliser values on the same scales and label both series so their changes can be compared directly.

Discrete categories do not become continuous merely because points can be joined. A line between time points shows an ordered change; a bar chart is usually clearer for unrelated categories.

For a linear graph, slope measures the change in $y$ per unit change in $x$ , and the vertical intercept is the value of $y$ where the line crosses the $y$ -axis at $x=0$ .

$slope m = change in y / change in x = (y₂-y₁)/(x₂-x₁); linear form y = mx + c$

Move	Example for $y=0.8x+0.2$
select two well-separated points on the straight line	$(1,1.0)$ and $(6,5.0)$
calculate vertical and horizontal changes	$\Delta y=4.0$ and $\Delta x=5$
divide and attach compound units	$m=4.0/5=0.8$ units of $y$ per unit of $x$
read or calculate the intercept	at $x=0$ , $y=c=0.2$
interpret	each one-unit rise in $x$ is associated with a 0.8-unit rise in $y$

Use points on the linear line, not necessarily two noisy raw points. The intercept may have no biological meaning when $x=0$ is impossible or lies outside the measured range; report it mathematically without inventing a biological claim.

Next Topic

—5 Geometry and trigonometryRead

4 Graphs

Learning objectives

Translate between a graph and its numerical evidence

Plot two variables on a graph that preserves the data

Find and interpret slope and intercept

Next Topic