Line Graphs for Scientists

Reminder:

For a straight-line graph, the equation is always of the format:

Therefore, if we have a straight-line graph, we can find the equation of the line by working out the gradient and the y-intercept.

We can use a spreadsheet to find a line of best fit and the relevant equation (see worksheet Lines of Best Fit, Scatter diagrams, further Confidence Intervals and Standard Deviation). However, this can also be done manually.

Positive gradients – line slopes upwards

Negative gradients – line slope downwards

New Material

You may do experiments for which the equation linking two variables is not of the format of a straight line! Sometimes the equation can be manipulated so that you can get the format of a straight line. You therefore need to work out what to plot on the axes to make a straight line and so be able to find the constants.

Here is a sample of some of the more common equations:

Equation / Changing to
y = mx+c
format / What to plot on y-axis / What to plot on x-axis / What is the Gradient / What is the y-intercept
R = kB2
R = rate
B = Concentration of reagent
k = a constant / R = kB2 + 0 / R / B2 / k / 0
L = a
L = Leak rate
d = depth
a = a constant / L = a+0 / L / / a / 0
y = ax2 + bx
y = Amount of bacteria
x= time (hours)
ab = constants / / / x / a / b

I = Light Intensity
d = distance from source
a = a constant / /
I
/ / a / 0
N = ent
N = Number of e-coli bacteria
t = time in minutes
e = exp. (2.7182818284….)
n = a constant / ln N = nt + 0 / ln N / t / n / 0
Y=axn
Y=Gate voltage
x= Drain current
an = constants / log y = n log x + log a / log y / log x / n / log a
(find antilog of value to work out a)

Questions:

1) The following readings were taken of the volume of a gas kept at atmospheric pressure whilst its temperature was varied. It is suspected that the model for this gas would fit the equation V = aT + b. Plot V against T (V on the y-axis, T on the x axis) and hence work out a and b

Volume V (cm3) / 190 / 205 / 220 / 235 / 250 / 265
Temperature T (oC) / 0 / 20 / 40 / 60 / 80 / 100

2) Below are the stopping distances for cars travelling at different speeds as shown in the Highway Code:

Speed (mph) / Stopping Distance (m)
20 / 12
30 / 23
40 / 36
50 / 43
60 / 73
70 / 96

i) Plot d against v2, where d metres is the stopping distance and v mph is the speed.

ii) Draw a straight line through the points

iii) Find the values of the gradient, m, and the intercept with the vertical axis, c, for your line.

iv) The quadratic model of the data is d = mv2 + c. Substitute your values of m and c and check that this closely models the data in the table.

3) The table below shows the amount of radiation, in milli-roentgens per hour, measured at various distances from a source of radioactivity:

Distance from source
( d ) (metres) / Radiation ( R ) (milli-roentgens per hour)
3.1 / 2100
6.5 / 500
8.6 / 270
9.2 / 240
13.2 / 110
18.5 / 60
30.8 / 20
33.8 / 17
36.9 / 15
43.1 / 11
49.2 / 8
61.5 / 5
70.8 / 4

a) Test whether the function R = fits the data by plotting R against d2 to see if this gives a straight line.

b) Draw an approximate line of best fit and find its equation. Hence find the value of k in the equation R =

4) The graph below shows the relation for PV = RT. What is the gradient of the graph?

FGTPage 1 of 4WorkFGDT 