$$y = 3x^2 + 1 \quad\Rightarrow\quad Y = 3X + 1$$

Let $X=x^2$, $Y=y$: the curve becomes a line on $X$–$Y$ axes. Reflex: Rename the variables, plot the renamed pair, and even a curve becomes a straight line.

$$Y = mX + c$$

Y, X are the new plotted variables; m is the gradient, c is the Y-intercept of the transformed line. Reflex: Apply all straight-line rules (gradient, intercept, substitution) to X and Y, not x and y.

$$y = 3x^2 + x \ \xrightarrow{\ \div x\ } \ \frac{y}{x} = 3x + 1$$

Manipulate (divide, multiply, take logs) until one term is a constant — that constant is $c$; the term still carrying the variable is $mX$.

Reflex: The constant term is $c$; the variable term is $mX$.

$$y^2 = 4x - 2$$

Already $Y = mX + c$: let $Y = y^2$, $X = x$, so $m = 4$, $c = -2$. Reflex: Check if it's already linear in renamed variables before manipulating anything.

$$\begin{gathered}y = 2e^x \Rightarrow \ln y = x + \ln 2\\[3pt] y = ax^n \Rightarrow \lg y = n\lg x + \lg a\end{gathered}$$

$y = 2e^x$: plot $\ln y$ against $x$ — $m = 1$, $c = \ln 2$. $y = ax^n$: plot $\lg y$ against $\lg x$ — $m = n$, $c = \lg a$. Reflex: Exponential in $x$ → plot $\ln y$ against $x$; power of $x$ → plot $\lg y$ against $\lg x$.

$$\begin{gathered}\frac{y}{x} = -2x^2 + 8\\[2pt] \implies\ y = -2x^3 + 8x\end{gathered}$$

Name $Y$ and $X$ from the axes, find the line's equation from its points, then un-rename and rearrange for $y$. Reflex: Renaming's just convenience — never forget what $Y$ and $X$ really are.

$$(X, Y) \text{ — never the raw } (x, y)$$

Points on the plotted line live in $X$-$Y$ space: convert $x \to X$ first, then find $Y$ from the line's equation. Reflex: Always answer with $(X, Y)$, never raw $(x, y)$.

Transform equation → tabulate new variables → plot best-fit line → read gradient & intercept → convert to constants. Off-line point flags bad data; read corrected value off the line. Related equations: draw auxiliary line, intersect.

Reflex: Transform → tabulate → plot → read m, c → convert to constants — graph answers carry a tolerance.