Converting to Linear Form

A relationship like $y = ax^2 + b$ doesn’t plot straight against $x$, but it plots perfectly straight against $x^2$. Converting to linear form means choosing new axes so the model becomes $Y = mX + c$, reading $m$ and $c$ from the line, and translating back to the original constants.

The matching method

Force the relationship into “(something) = constant $\times$ (something else) + constant”:

Model	Plot $Y$ against $X$	Gradient	Intercept
$y = ax^2 + b$	$y$ vs $x^2$	$a$	$b$
$y = \dfrac{a}{x} + b$	$y$ vs $\dfrac{1}{x}$	$a$	$b$
$y = ax + \dfrac{b}{x} \to xy = ax^2 + b$	$xy$ vs $x^2$	$a$	$b$
$\dfrac{y}{x} = ax + b$	$\dfrac{y}{x}$ vs $x$	$a$	$b$
$\sqrt{y} = ax + b$	$\sqrt{y}$ vs $x$	$a$	$b$

The technique: divide, multiply or substitute until each variable appears in exactly one compound, then say what’s plotted against what. That statement. “plotting $xy$ against $x^2$ gives a straight line with gradient $a$ and intercept $b$”, is itself a mark.

Data follows $y = ax + \dfrac{b}{x}$. Linearise. Multiply by $x$: $xy = ax^2 + b \to$ plot $xy$ against $x^2$; gradient $a$, intercept $b$.

Recovering the constants

From the drawn line (or a given gradient/intercept): compute the gradient from two well-separated points on the line, read the intercept, and translate back. Two cautions: the intercept must be read where $X = 0$ (which is $x^2 = 0$, not necessarily the left edge of the printed graph), and any compound like $\sqrt{y}$ or $xy$ must be undone when reporting, finish by writing the original model with numbers: $y = 2.5x + \dfrac{7}{x}$. The log-based versions ($y = ax^n$, $y = Ab^x$) follow the same script with an extra unlog step, covered here.

Given data tables, build the transformed columns ($x^2$, $xy$, …) first, exactly, these feed the gradient, and transcription slips propagate.

Common mistakes

• Plot variables chosen so a variable appears on both axes in solvable form (no valid match shown)
• Intercept read at the graph’s left edge instead of $X = 0$
• Compound variables never undone ($\sqrt{y}$ reported as $y$)
• Gradient from two adjacent points instead of the line’s span
• The final model never assembled with its numerical constants

Full topic context: Straight-Line Graphs notes · the log version: reducing to linear form.