Graphs (M3)
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Many graphs have on the horizontal axis and on the vertical axis, but in a scientific context, other letters are sometimes used for the two variables.
When writing down an equation of a line or curve, use the correct letters as they appear on the graph.
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.
If and have a reciprocal relationship, it means that for some number . We can rearrange the equation:
Since division by zero is impossible, takes no value when , so the graph has no -intercept. In fact, the graph will not touch either axis.
If is large, then must be small, and vice versa, so the graph is very flat for large but very steep for close to zero.
You should be able to recognise reciprocal graphs by noticing these two properties.
You need to be able to recognise exponential graphs, which can show:
- exponential growth, for example, a quantity that quadruples every hour
- exponential decay, for example, a quantity that halves every second.
Examples of exponential graphs include:
- shows exponential growth, and the graph grows rapidly.
- shows exponential decay, and the graph decreases towards zero without ever crossing the X axis.
In both cases, is the value of the x-intercept, because when ,
Here, the green graph decays towards zero, while the red graph grows exponentially.
A quadratic graph has the shape of a parabola, resembling the letter U.
The equation of a quadratic graph is where and represent numbers that define the precise shape and position of the graph.
It is important to note:
- is the value of the -intercept
- If is made negative, the graph opens downwards; rather than a U-shape, it is upside-down like ∩.
It is useful to know:
- if is large (positive or negative), then the parabola will be narrower
- if is positive, the graph is sloping upwards at the -intercept. If is negative, the graph is sloping downwards at this point.
- This parabola opens downwards, which tells you that the coefficient of is negative.
- The graph is not very steep, which indicates that is not large. In fact,
- The -intercept is which tells you that the constant term,
- At the -intercept, the graph slopes upwards, so we know that In fact,
A straight line through the origin tells you that is proportional to
Sometimes you will see a graph that starts at the origin and curves upwards. Graphs like this tell you that is proportional to a power of
This can be written as for some number It can also be written as where is some constant of proportionality.
An example of a power relationship is shown in the graph above. The equation of the line is where is proportional to and the constant of proportionality is
It is challenging to determine these exact values solely by examining the graph. Luckily, you won’t need to do that!
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.
To make your graphs easy to read, it is important to choose suitable scales for both axes.
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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.
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.
You should also be aware that velocity (or speed) is the rate of change of position, and acceleration is the rate of change of velocity (or speed).
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.
It is important to note that the word per can be replaced by a forward slash (/). Alternatively, you can raise the quantity on the horizontal axis to the power of negative one. Both alternatives are shown in the parentheses above.
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.
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.
It can be difficult to draw the tangent line accurately! It should touch the curve without crossing it.
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.
You may be asked to find the instantaneous rate of change at a particular moment, or the average rate of change over a particular time interval.
It is important that you calculate the correct rate!
The example graph above shows a vehicle that travelled 30 metres in 10 seconds. Its average velocity was therefore over this interval.
However, at four seconds, the graph is horizontal, showing that the vehicle was not moving. At this moment, its instantaneous velocity was
Moreover, at eight seconds, the instantaneous velocity was about as illustrated by the green tangent line.
Question walkthrough
Calculate an average rate of change
Determining the average rate of change of height from a graph.
Speed and velocity each describe the rate of change of position.
However, there is one important difference:
- Speed is a scalar quantity; it is not affected by the direction of motion.
- Velocity is a vector quantity. It is positive in the forward direction and negative in the backwards direction.
When considering velocity, consider whether the motion is forward or backwards.
An example of this, is if we regard movement towards the right to be positive, when the car is moving in the opposite direction:
- its speed is positive, (e.g. )
- its velocity is negative, (e.g. )
Recall:
- Velocity is the rate of change of displacement (rate of change of position).
- Acceleration is the rate of change of velocity.
These are vector quantities, so they can be positive or negative depending on the motion.
The area under a graph sometimes has significance, although not always! You should be aware of the following examples:
When calculating a quantity using the area under a graph, any areas beneath the horizontal axis count as negative. You should find the areas above the axis and subtract any areas beneath it.
Consider the following velocity–time graph. How would you calculate the total displacement after seconds?
When finding the area under a graph by counting rectangles, your answer will be an estimate.
To get a good estimate, use the following rule:
- If more than half of the rectangle is under the curve, include it.
- If less than half of the rectangle is under the curve, don’t include it.
- If you are not sure, because it is approximately half under the curve, count it as one-half.
You can approximate the area under this curve by counting the squares:
Question walkthrough
Use geometry to find the area under a graph.
Uses the area under a velocity-time graph, split into a trapezium and a triangle, to find the distance of a town from the start of a train's journey, then interprets a negative area from a change in direction.
Question walkthrough
Count rectangles to estimate the area under a graph.
Uses the area under a charge-voltage graph to estimate the total energy discharged by a smartphone battery, by counting grid rectangles and multiplying by the energy each one represents.
The area under a graph can be calculated using graphical methods, like counting squares.
It is useful to note that the area can also be determined using calculus, although you won’t need to do this in your Science exams.
If you are studying A-level Mathematics, you may have seen that differentiation is a technique for finding the gradient function of a graph. The opposite is antidifferentiation or integration.
The Fundamental Theorem of Calculus states that antidifferentiation of a function is the same as finding the area under the graph of that function.
In the image, the area under could be calculated either:
- by graphical methods; or
- by calculating the antiderivative and substituting in values for
Some quantities (such as the number of radioactive particles in a large sample) decrease by the same proportion in each time interval. For example, the quantity might decrease by 20% each second, so that 80% remains.
The rate of decrease does not depend on previous values, only on its current value. Thus, it is sometimes described as memoryless.
Memoryless decay is modelled using an exponential curve, as shown.
It always has the equation:
Where:
- is the initial value (as shown), and
- is the decay constant.
The greater the decay constant, the more rapid the decay.
Some quantities (such as radioactive activity) decrease by the same proportion in each time interval.
This can be modelled by the differential equation:
Where:
- is the decay constant, which depends on the rate of decay.
In formal notation, this can be written:
This always has the solution:
Where is the initial value.
Observe in the graph above that the gradient is always proportional to the value of as expected!
The decay constant appears in the equation for exponential decay:
It also appears in the rate equation – a related formula that describes decay:
It is useful to know that a larger decay constant means the decay is more rapid.
As you can see, these exponential curves have a similar shape. But the green curve with a decay constant of 0.1 decays very slowly. The orange curve with a decay constant of 2 decays much more rapidly.
Question walkthrough
Apply understanding of exponential curves to solve a rate equation.
Uses the standard solution to an exponential decay rate equation to find the remaining mass of a radioactive isotope after a given time, using the initial mass to find the constant of integration.
You may encounter graphs where the vertical axis shows logarithms of a measurement. In some cases, the horizontal axis may also show logarithms. These are both logarithmic plots.
These graphs often show a straight line. However, this doesn’t mean they represent a linear relationship. You must not confuse them with a linear relationship.
You may need to interpret a semi-log graph, where the vertical axis shows a logarithm while the horizontal axis does not.
If this graph shows a straight line, this always indicates an exponential curve of the form where is the slope or gradient of the straight line.
To see this, take the equation of the straight line, and take the exponent of both sides:
You may need to interpret a log–log graph with both axes showing logarithms.
If the graph shows a straight line, this always indicates a power law of the form where is the slope or gradient of the straight line.
To see this, take the equation of the straight line, and take the exponent of both sides:
Question walkthrough
Determine an exponential relationship from a semi-log graph.
Uses a semi-log graph of a capacitor's discharge voltage to find the initial voltage and decay constant, then uses the decay constant and resistance to calculate the capacitance.
Question walkthrough
Determine a power law relationship from a log–log graph.
Uses a line of best fit on a log-log graph to find the gradient and intercept, then converts these into a power-law equation relating y and x.
The sine wave models many relationships in Physics, including circular motion and simple harmonic motion. You need to recognise the graph of and be aware of its key properties.
It is important to note the following features of
- Its maximum value is 1, and its minimum value is –1.
- It passes through the origin; .
- It is periodic (repeating every radians or 360 degrees).
- It is odd (the negative part of the graph is the same as the positive part, but rotated).
The sine wave models many relationships in Physics, including circular motion and simple harmonic motion. One form of the sine wave is the cosine, written as You need to recognise this graph and be aware of its key properties.
It is important to note the following features of
- Its maximum value is 1, and its minimum value is –1.
- It has its maximum value when In other words, .
- It is periodic (repeating every radians or 360 degrees).
- It is an even function (the negative part of the graph is the same as the positive part, but reflected).
You will often encounter graphs of and Each of these is also a sine wave, but there are some key differences.
It is important to note the following features of these graphs:
- As the expressions are squared, they only take positive values, from a minimum of 0 to a maximum of 1.
- passes through the origin like .
- has its maximum value when like .
- They are periodic, but with double the frequency, repeating every radians or 180 degrees. This is because the negative “half” of the wave becomes an extra (positive) wave.
- and are even functions like . The negative part of the graph is the same as the positive part, but reflected.
The tan graph is an important shape in physics and mathematics. You need to recognise the graph and be aware of its key properties.
It is important to note the following features of
- It passes through the origin; .
- It is periodic, repeating every radians or 180 degrees.
- It is an odd function (the negative part of the graph is the same as the positive part, but rotated).
- It has a vertical asymptote at etc. meaning that the graph becomes arbitrarily high or low as approaches these values.
It is useful to know that the vertical asymptote arises because a right-angled triangle with an adjacent side of length 1 could have an opposite side arbitrarily large. Another way to see this is using the identity:
When this fraction means a division by a number close to zero.