Recognise non-linear association from a scatterplot and residual plot, apply the squared, logarithmic or reciprocal transformation to the explanatory or response variable to linearise the data, fit a least-squares line to the transformed data, and use it to predict

What this dot point is asking

VCAA wants you to handle bivariate data whose scatterplot is curved rather than linear. A least-squares line should only be fitted to a linear relationship, so when the residual plot shows a clear pattern you first transform one of the variables, using a squared, logarithmic or reciprocal transformation, to straighten the data. You then fit the least-squares line to the transformed data and use it to predict, remembering to undo the transformation at the end. This is the natural follow-on from correlation and regression.

Spotting that a transformation is needed

A single curved scatterplot, or a residual plot with a clear arch or U-shape rather than random scatter, signals that a straight line is the wrong model. Rather than abandon regression, you re-express one variable so that the relationship becomes linear.

Choosing the transformation

The direction the curve bends tells you which transformation to apply. Stretching the high end of the $x$-axis (squaring $x$) or compressing it (log or reciprocal of $x$) shifts points to straighten the bulge. In the exam you are usually told which transformation to apply, or you pick the one that gives the better $r^2$ on the transformed data.

Fitting and predicting with a transformed model

Once a variable is transformed, treat the transformed quantity as a new variable and fit the least-squares line as normal.

Reading the transformed equation back

The fitted equation already contains the transformation, so prediction is just careful substitution. If the transformation was on the response variable, for example $\sqrt{y} = a + bx$ written as $y^{1/2}$ or with $y$ replaced by $\log y$, you must undo it at the end: square both sides, or raise $10$ to the power. Always check whether the transformation sits on $x$ or on $y$ before predicting.

The circle of transformations idea

A helpful way to choose a transformation is to picture the curve's bulge. If the scatter bulges so that the curve rises steeply then flattens (concave down), a log or square-root style re-expression of $x$, or squaring $y$, tends to straighten it. If the curve rises gently then steepens (concave up), squaring $x$, or taking a log or reciprocal of $y$, often works. The six VCE moves are $x^2$, $\log_{10}x$, $\tfrac{1}{x}$ and the same three on $y$. Each shifts points along one axis: squaring stretches the high end, while log and reciprocal compress it. You rarely need to derive the choice from scratch in the exam, because the question usually names the transformation, but understanding why it works helps you avoid applying it to the wrong variable.

Comparing $r^2$ before and after

The justification for a transformation is that it improves the linearity of the relationship, which shows up as a higher coefficient of determination $r^2$ and a residual plot that becomes random scatter. A standard exam task is to fit the line to the raw data, note a patterned residual plot and a modest $r^2$, then apply the named transformation, refit, and report the new $r^2$ as evidence that the transformed model is better. Always state the improvement with figures, for example that $r^2$ rises from $0.78$ to $0.96$, and confirm that the residual plot of the transformed model shows no remaining pattern. That combination of a higher $r^2$ and a structureless residual plot is the full justification markers want.

Predicting and the limits of the model

Once you have the transformed equation, prediction is careful substitution: transform the input if $x$ was transformed, substitute, then undo any transformation on $y$ at the very end. The same warnings as ordinary regression still apply, only more sharply. Extrapolating a transformed model far beyond the data range is even more unreliable than for a linear model, because the chosen transformation was only justified over the observed range. State that a prediction is an interpolation (reasonably reliable) or an extrapolation (treat with caution), and never read a transformation as proof that one variable causes the other.

Why this matters for the exams

Transformation questions appear most years and reward students who keep track of which variable was transformed and who undo it correctly when predicting. They build directly on correlation and least-squares regression: the residual plot is the trigger, the transformation is the fix, and the prediction is the payoff. Show the transformed value explicitly in your working so a marker can follow each step.