Linearising a non-linear relationship

Key ideas

A non-linear relationship (e.g. a curve on a graph of $y$ against $x$ ) is hard to analyse directly — it's difficult to read an exact mathematical rule off a curve.
Linearising means re-plotting the data so the graph becomes a straight line. A straight-line graph can be analysed precisely using its gradient and y-intercept.
The general straight-line equation is:

$y = mx + c$

$m$ is the gradient.
$c$ is the y-intercept.
To linearise, you plot a transformed version of one or both axes — chosen based on the shape you suspect the relationship has.

Common linearising transformations

Suspected relationship	Plot this on the vertical axis	Against	Straight-line form
$y = kx^2$	$y$	$x^2$	gradient $= k$
$y = k\sqrt{x}$	$y$	$\sqrt{x}$	gradient $= k$
$y = \dfrac{k}{x}$	$y$	$\dfrac{1}{x}$	gradient $= k$
$y = kx^n$ (unknown power $n$ )	$\log y$	$\log x$	gradient $= n$ , intercept gives $\log k$

If you don't know the power in advance, a log–log plot ( $\log y$ vs $\log x$ ) is the general tool: taking logs of $y = kx^n$ gives

$\log y = n\log x + \log k$

which is a straight line with gradient $n$ (the power) and y-intercept $\log k$ (so $k = 10^{\text{intercept}}$ ).

Worked ExampleLinearising a pendulum investigation

A student suspects the period $T$ of a pendulum depends on its length $L$ as $T = kL^n$ . Explain how to use a log–log plot to find $n$ and $k$ .

Step 1 — Take logs of both sides

Starting from $T = kL^n$ , taking $\log$ of both sides gives:

$\log T = n\log L + \log k$

Step 2 — Match to the straight-line equation

Comparing with $y = mx + c$ : plotting $\log T$ (vertical axis) against $\log L$ (horizontal axis) gives a straight line with

$\text{gradient } m = n, \qquad \text{intercept } c = \log k$

Step 3 — Find $n$ and $k$ from the graph

Draw a line of best fit through the $\log T$ vs $\log L$ points. The gradient of that line IS the power $n$ . Reading the y-intercept $c$ off the graph, $k$ is found by "undoing" the log:

$k = 10^{c}$

Practice question

A student plots $\log(\text{bounce height})$ against $\log(\text{drop height})$ and gets a straight line with gradient $1.0$ and y-intercept $-0.30$ . State the relationship between bounce height $h$ and drop height $H$ .

Worked solution: Gradient $= n = 1.0$ , so $h = kH^{1.0} = kH$ (bounce height is directly proportional to drop height). $k = 10^{-0.30} \approx 0.50$ , so $h \approx 0.50H$ — the ball bounces back to about half the height it was dropped from.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A student suspects $y = kx^2$ . State what should be plotted on each axis to produce a straight line, and what the gradient of that line represents.

Merit

A log–log plot of resistance $R$ against wire length $l$ gives a straight line with gradient $1.0$ . Explain what this shows about the relationship between $R$ and $l$ .

Excellence

A student plots raw bounce height $h$ against drop height $H$ and gets a curve that looks almost straight near the origin. They conclude the relationship is linear without linearising. Evaluate this conclusion.

Linearising a non-linear relationship

Covers AS91168

Key ideas

A non-linear relationship (e.g. a curve on a graph of $y$ against $x$ ) is hard to analyse directly — it's difficult to read an exact mathematical rule off a curve.
Linearising means re-plotting the data so the graph becomes a straight line. A straight-line graph can be analysed precisely using its gradient and y-intercept.
The general straight-line equation is:

$y = mx + c$

$m$ is the gradient.
$c$ is the y-intercept.
To linearise, you plot a transformed version of one or both axes — chosen based on the shape you suspect the relationship has.

Common linearising transformations

Suspected relationship	Plot this on the vertical axis	Against	Straight-line form
$y = kx^2$	$y$	$x^2$	gradient $= k$
$y = k\sqrt{x}$	$y$	$\sqrt{x}$	gradient $= k$
$y = \dfrac{k}{x}$	$y$	$\dfrac{1}{x}$	gradient $= k$
$y = kx^n$ (unknown power $n$ )	$\log y$	$\log x$	gradient $= n$ , intercept gives $\log k$

If you don't know the power in advance, a log–log plot ( $\log y$ vs $\log x$ ) is the general tool: taking logs of $y = kx^n$ gives

$\log y = n\log x + \log k$

which is a straight line with gradient $n$ (the power) and y-intercept $\log k$ (so $k = 10^{\text{intercept}}$ ).

Worked ExampleLinearising a pendulum investigation

A student suspects the period $T$ of a pendulum depends on its length $L$ as $T = kL^n$ . Explain how to use a log–log plot to find $n$ and $k$ .

Step 1 — Take logs of both sides

Starting from $T = kL^n$ , taking $\log$ of both sides gives:

$\log T = n\log L + \log k$

Step 2 — Match to the straight-line equation

Comparing with $y = mx + c$ : plotting $\log T$ (vertical axis) against $\log L$ (horizontal axis) gives a straight line with

$\text{gradient } m = n, \qquad \text{intercept } c = \log k$

Step 3 — Find $n$ and $k$ from the graph

Draw a line of best fit through the $\log T$ vs $\log L$ points. The gradient of that line IS the power $n$ . Reading the y-intercept $c$ off the graph, $k$ is found by "undoing" the log:

$k = 10^{c}$

Practice question

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A student suspects $y = kx^2$ . State what should be plotted on each axis to produce a straight line, and what the gradient of that line represents.

Merit

A log–log plot of resistance $R$ against wire length $l$ gives a straight line with gradient $1.0$ . Explain what this shows about the relationship between $R$ and $l$ .

Excellence

Linearising a non-linear relationship

Key ideas

Common linearising transformations

Step 1 — Take logs of both sides

Step 2 — Match to the straight-line equation

Step 3 — Find nnn and kkk from the graph

Practice question

Test yourself

Practice by grade

Linearising a non-linear relationship

Key ideas

Common linearising transformations

Step 1 — Take logs of both sides

Step 2 — Match to the straight-line equation

Step 3 — Find nnn and kkk from the graph

Practice question

Test yourself

Practice by grade

Step 3 — Find $n$ and $k$ from the graph

Step 3 — Find $n$ and $k$ from the graph