11.
The power functions
The family of functions
y
= axb,
called the power functions, are very powerful! They include simple quadratic
(b = 2), cubic (b =3), square root (b = 0,5), and hyperbola (b = -1) functions,
and are therefore very general.
Open the Power Function
applet below. Click on the sliders to change the parameters a and b over
a wide range, i.e. positive and negative, larger than 1, smaller than
1, etc. Also note what happens when b is exactly a whole number, e.g.
b = -1, 0, 1 or 2. Can you explain why this happens?
Linearisation
Linearising a power function to recognise it, involves an interesting
application of logarithmic concepts. First open the Excel Power tool to
investigate the relationship numerically, then read the algebraic explanation:
|
In 1619,
Johannes Kepler published his third law of planetary motion,
which related the
period
and the mean mean distance of
the then six known planets.
Planet |
Distance
(AU) |
Period
(years) |
Saturn |
9,510 |
29,4571 |
Jupiter |
5,200 |
11,8621 |
Mars |
1,524 |
1,8809 |
Earth |
1,000 |
1,0000 |
Venus |
0,724 |
0,6152 |
Mercury |
0,388 |
0,2408 |
Use the
Scatterplot applet (Load Kepler then and Kepler now) and this Excel tool
to try to recreate Kepler’s third law of planetary motion.
|
|
|
The table shows experimental
data about the relationship between the length of a
pendulum in centimetres and its period in seconds (the time to complete one oscillation).
Find
the relationship between the length and period by linearising
the data in Pendulum in Scatterplot.
Then
use the Excel pendulum to fit a power function
on the data.
|
|
Length |
Period
(s) |
|
6,5 |
0,51 |
|
11,0 |
0,67 |
|
13,2 |
0,73 |
|
15,0 |
0,79 |
|
18,1 |
0,89 |
|
23,0 |
0,98 |
|
24,4 |
1,01 |
|
26,5 |
1,08 |
|
30,6 |
1,13 |
|
34,3 |
1,25 |
|
37,5 |
1,28 |
|
41,5 |
1,33 |
|
|
|
|
Let’s
revisit two previous activities, but this time fit a power
function to the data:
|
1. |
Revisit
the Trout problem:
Use
the data to find an algebraic model of the relationship
between length and weight of trout, and use the model
to estimate the weight of a 20 cm and of a 120 cm trout
fish.
You
can use the scatterplot (in the drop-down list, scroll
down to Trout, load it, and update the display).
You can also use this Excel worksheet.
|
| Length
(x
cm) |
Weight
(y
kg) |
| 27 |
0,22 |
| 42 |
0,85 |
| 46 |
1,12 |
| 15,0 |
0,79 |
| 54,5 |
1,86 |
| 60 |
2,49 |
| 68 |
3,63 |
| 72,5 |
4,4 |
| 85 |
7,09 |
|
2.
|
Revisit the
Boyle problem:
Fit a power function to the data to deduce the relationship
between the pressure and volume of a gas.
Click
here for a discussion:
|
| Pressure
(x)
|
Volume
(y) |
| 0,5 |
21 |
| 0,56 |
19 |
| 0,63 |
17 |
| 0,71 |
15 |
| 0,83 |
13 |
| 0,99 |
11 |
| 1,09 |
10 |
| 1,21 |
9 |
| 1,37 |
8 |
1,58 |
7 |
1,83 |
6 |
2,2 |
5 |
2,72 |
4 |
|
|
|
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