Graphs (M3)
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A relationship between two quantities is proportional if one quantity is simply a multiple of the other. Any graph of this would show a straight line through the origin.
The equation of this graph would be , where is the gradient of the line and is also the constant of proportionality. For example, in the green and blue lines below the gradient is .
Graphs of the form are also straight lines. But unless the -intercept is zero, the line does not pass through the origin, and the graph does not show proportionality.
Most straight-line graphs should be written in the form where is the gradient.
If the gradient is zero, it means that remains constant: it never changes.
In this case, which is better written as Keeping things simple with equations makes it easier to check your work and more quickly wrap your head around what comes next.
Graphs are for communicating information to other people, so clear labelling is crucial.
Always remember to include a brief title for the graph, and axis labels (including units) to clarify what measurements the data represents.
This graph shows the approximate volume of gas produced two minutes after the experiment started.
This graph has a clear title and axis labels, which allow us to answer questions like this confidently.
It is important to note that you may lose marks in your exam if you draw a graph that omits these features.
When plotting a graph, the axes are determined by the variables in an experiment.
- x-axis – independent variable (the thing that is changed)
- y-axis – dependent variable (the thing that is measured).
To make your graphs easy to read, it is important to choose suitable scales for both axes.
To make your graphs easy to read, it is important to choose suitable scales for both axes. It is usually best to use multiples of values related to British coins and notes, such as 1, 10, 100, …, 2, 20, 200, …, 5, 50, 500.
Lines of best fit (and curves) usually have a similar number of data points above the line (or curve) as below it. If you draw a line of best fit for which this is not true, you should reconsider whether it is accurate!
It is useful to know that in very few extreme cases, the number of data points above and below the line of best fit could be quite different. But this is rare!
You must draw lines of best fit accurately, as you may need to read precise values from them. Therefore, it is important to use a ruler or other straight edge.
Make sure you bring one to your exam!
For data that follows a linear trend, you should draw a straight line of best fit through the data, using a ruler.
Never join the data dot-to-dot unless you are sure that the trend is not linear. For example, monthly climate data does not follow a linear trend.
It is important to note that exam questions may ask you to interpolate, estimate, or predict values from a graph that shows a clear trend.
You should do this by drawing a line of best fit (or the curve) and reading values from that, rather than reading individual data points.
This is because the data points will fall above or below the line (or curve) due to unrelated factors (known as “noise”), such as human error. The line (or curve) reduces the effect of this noise.
Question walkthrough
Draw a line of best fit to extrapolate from data.
Draws a line of best fit through weight-extension data for an elastic fibre, then extends it to the y-axis to find its natural length when no weight is attached.
Question walkthrough
Draw a curve to extrapolate from data.
Draws a smooth curve of best fit through non-linear data, then uses it to extrapolate a prediction for lemonade sales at a forecast temperature.
In some scientific contexts, it is clear that any line of best fit (or curve) must pass through the origin. Some examples of where the graph must pass through the origin:
- In Biology and Chemistry, in rate–concentration graphs, because when the concentration of the reactant is zero, the reaction does not happen.
- In Physics, stress–strain graphs, because when no force is applied to the material, the material has zero deformation.
- In any context, when a quantity has an initial value of zero.
You should use your scientific understanding to identify these cases and ensure that any line of best fit (or curve) passes through the origin.
It is important to note that in many other scenarios, there is no reason to expect that the line (or curve) would pass through the origin.
For the data in this graph, which of the two lines should you use? It depends on the scientific context. You will need to choose!
Where a straight-line graph crosses an axis, the crossing point is called an intercept.
The point where the graph crosses the vertical y-axis is called the y-intercept. This is often given the value when writing the equation of the straight line:
The point where the graph crosses the horizontal x-axis is called the x-intercept.
Here, the y-intercept is 3, and the x-intercept is 6.
Question walkthrough
Understand that y = mx + c represents a linear relationship
Translating an equation from numerical form to graphical form.
To calculate the gradient of a graph, choose any two points and calculate:
If the two points are and the same expression can be written:
However, your answer may be inaccurate if you divide by a horizontal change that is too small. To avoid this, choose two points that are quite far apart.
If a straight-line graph is sloping downwards, the gradient is negative.
You should use the same procedure to calculate the gradient, but make sure that the vertical change is negative because it’s a decrease.
Sometimes you will be shown a graph of data from an experiment. The points follow a linear trend, and a (straight) line of best fit may be used to illustrate the trend.
When estimating the slope and intercept of this graph, use the line itself, not the individual data points.
On most graphs, the horizontal axis starts at and therefore the y-intercept is on the vertical axis.
However, sometimes the horizontal axis is drawn differently. In these cases, you would need to do some further calculations to determine the y-intercept.
Question walkthrough
Determine the slope and intercept of a linear graph
Using a drawn slope to determine a derived quantity from a linear graph.
You may be asked to find the gradient of a straight-line graph, which is also known as the slope of the graph.
The gradient illustrates a rate of change, so some exam questions may use phrases similar to this instead. You need to recognise phrases about rates and interpret them as the gradient of a graph, if relevant.
For example, a question might show a graph of concentration against time, and ask about the rate of increase or rate of decrease of the concentration.
If a straight-line graph has a positive gradient (it slopes upwards), then it shows a rate of increase.
If a straight-line graph has a negative gradient (it slopes downwards), then it shows a rate of decrease.
If the gradient of the straight line is zero (it is horizontal), then it shows that the quantity is not changing – it is constant.
The green line has a gradient of 0.1. It shows an increase of 0.1 units per second.
The blue line has a gradient of zero. It shows that the quantity is constant.
The red line has a gradient of −0.06. It shows a decrease of 0.06 units per second.
The units for a rate of change can be written in words by arranging the units from the axes:
Examples of this include:
- kilometres per hour (or or )
- cubic centimetres per second (or or )
- Euros per kilogram (or € or €)
As noted in the examples, you may abbreviate these units in various ways to reduce the amount you need to write.
Exam boards generally accept both formats because they’re both scientifically correct ways to express the same thing. The key is being consistent and clear in your notation.
On a straight-line graph, the gradient is the same everywhere. The graph describes a quantity that always changes at the same rate.
On a curve, the gradient is changing. The curve doesn’t rise or fall evenly; it changes based on the X-axis value.
Sometimes, your first attempt at drawing a tangent line may not be accurate. For example, the tangent line could cross the curve.
If that happens, you should erase the inaccurate tangent line and try again.
You should use a pencil so that it is convenient to erase a line if necessary.
It can be difficult to draw the tangent line accurately! It should touch the curve without crossing it.
Question walkthrough
Draw and use the slope of a tangent to a curve as a measure of rate of change
Uses a tangent to a curve to estimate the instantaneous speed of a machinery component at a given time, by drawing the tangent and measuring its gradient between two points.