Sketch is an instruction, not a lesser draw

what most people hand in

1 of 3marks earned

nonefeatures labelled

not enoughverdict

The same curve each time. What changes is what is written on it.

The figure needs JavaScript. It shows y = x³ − 3x three ways: an unlabelled curve, which earns 1 mark of 3; a sketch with axes, the roots at 0 and ±1.73, and the turning points (−1, 2) and (1, −2) all labelled, which earns 3; and an accurate draw on a grid with points plotted. Everything below stands on its own.

What each word asks for

	Sketch	Draw
Accuracy	Not to scale. The shape and the relationships have to be right.	To scale, with points plotted and a ruler for any straight line.
Grid	Not needed. Plain paper is fine.	Expected, and the scale has to be stated.
Labels	Axes, and every key feature the question cares about.	Axes, and the scale on each.
How long	A minute, done properly.	Several minutes. If a question wants a draw it will give you a grid.

Neither word means "rough". A sketch is a specification: the right shape, in the right place, with the right things named. It is faster than a draw because it does not need accuracy, not because it needs less thought.

What counts as a key feature

For y = x³ − 3x, a sketch that earns everything shows:

Both axes labelled, x and y. One mark is routinely lost here and it takes two seconds.
The x-intercepts. Here x³ − 3x = x(x² − 3), so the roots are 0 and ±1.73.
The y-intercept, which is 0, the same point.
The turning points. A local maximum at (−1, 2) and a local minimum at (1, −2).
The behaviour at the ends: down on the left, up on the right, which is what the positive x³ term does.

A local maximum is not the maximum. The point (−1, 2) is the top of a hill, not the top of the graph: y = 10 gives 970 and it keeps going. Label it "local maximum" or "local max". A student who writes "maximum" has made a false claim about a cubic, and an examiner reading that cannot award the feature.

Getting it off the screen

The syllabus asks specifically for transferring a graph from a screen to paper. Four things go wrong, in this order:

The window becomes the graph. Whatever the screen shows gets copied, including the fact that the curve appears to stop at the edges. It does not stop.
The scales are not copied. If the window ran −3 to 3 on x and −10 to 10 on y, the drawing is six units wide and twenty tall and the shape is not what you saw.
Features outside the window are missed. Always widen once and look before you copy.
Nothing is labelled, because the screen did not label it.

Sums and differences of functions. The syllabus expects you to graph f(x) + g(x) and f(x) − g(x) with technology, including functions that are not in this topic. There is nothing to learn: enter it as one expression. What is worth doing is predicting the shape first, then checking. For y = x³ − 3x, the x³ wins far from zero and the −3x wins near it, which is exactly why there are two turning points close to the origin and nothing else anywhere.

On the GDC: a window that shows the features

The default window hides features on almost every cubic, and a sketch copied from a window that hides them will be wrong in a way nobody can see from the drawing.

When you may use it. Applications allows a calculator in every paper. Analysis does not allow one in Paper 1, where a sketch has to come from the algebra: factorise for the roots, differentiate for the turning points.

TI-Nspire CX II

ctrl doc → Add Graphs, then x x² for a square, or type the cubic as x*x*x-3x to avoid the exponent box
menu → Window / Zoom → Window Settings, x from -3 to 3, y from -4 to 4
menu → Analyze Graph → Maximum, then set a left and a right bound around the hill: (-1, 2)
Repeat with Minimum for (1, -2), and Zero three times for the roots

Casio fx-CG50

MENU → Graph, enter Y1=X^3-3X Use the x² key or the right arrow to leave the exponent: typing ^ opens a superscript box and everything after it stays inside.
SHIFT F3 V-Window, Xmin -3, Xmax 3, Ymin -4, Ymax 4, then F6 DRAW
SHIFT F5 G-SOLVE → F2 MAX: (-1, 2)
Then MIN for (1, -2), and ROOT with the arrow keys for each of the three roots

The mark people lose. Copying the window instead of the graph. On the default window this cubic is a near-vertical line through the origin and both turning points are invisible, so a sketch made from it has no features to label and cannot score. Set the window deliberately, find the features with the machine, and then draw. And on the Casio, G-SOLVE is SHIFT F5, not F5.

Your turn

1. For y = x³ − 3x, give the largest root, to 2 decimal places.

2. What is the y-coordinate of the local maximum?

3. How many x-intercepts does it have?

4. A question says "sketch" and gives no grid. A student produces an accurate plot on squared paper with every point measured, and labels nothing. How many of the three marks?

5. Which label on (−1, 2) is correct?

Question 5. Which label on the point minus 1 comma 2 is correct? Maximum Local maximum Vertex Turning point, with no other word needed

Where the marks go

1 markThe correct shape, with the right end behaviour.

1 markThe intercepts, marked and valued.

1 markThe turning points, with coordinates and the word "local".

And axes labelled, which is not usually its own mark but is a condition on the others: an examiner cannot read a coordinate off an unlabelled pair of lines.