WAMath MethodsSyllabus dot point

How do we combine intercepts, stationary points, concavity and asymptotes into a complete and accurate sketch of a curve?

Sketch curves using intercepts, stationary points, their nature, points of inflection, asymptotes and end behaviour

WACE Year 12 Mathematics Methods Unit 3 curve sketching with calculus: a systematic method using intercepts, stationary points and their nature, points of inflection, asymptotes and end behaviour, with a fully worked SCSA-style sketch.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

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What this dot point is asking
A systematic method
Putting it together
Asymptotes for non-polynomial curves

What this dot point is asking

SCSA Unit 3 asks you to assemble the calculus tools into one organised process that produces an accurate graph. This dot point is examined in both the calculator-free and calculator-assumed sections; in the calculator-free section the sketch must come entirely from by-hand analysis.

A systematic method

Work through the same checklist every time so no feature is missed.

Putting it together

The features divide the $x$ -axis into intervals on which the curve is increasing or decreasing and concave up or down. Once the key points are plotted, the curve is drawn to match both the gradient sign (from $f'$ ) and the concavity (from $f''$ ) on each interval.

A full curve sketch

Sketch $f(x)=x^{3}-3x^{2}$

Intercepts. $f(0)=0$ , so the $y$ -intercept is $0$ . Setting $f(x)=0$ gives $x^{2}(x-3)=0$ , so $x$ -intercepts at $x=0$ (repeated) and $x=3$ .

Stationary points. $f'(x)=3x^{2}-6x=3x(x-2)=0$ , so $x=0$ and $x=2$ . The $y$ -values are $f(0)=0$ and $f(2)=8-12=-4$ .

Nature. $f''(x)=6x-6$ . At $x=0$ , $f''(0)=-6<0$ , a local maximum $(0,0)$ . At $x=2$ , $f''(2)=6>0$ , a local minimum $(2,-4)$ .

Inflection. $f''(x)=0$ at $x=1$ ; the sign of $f''$ changes from negative to positive, so $(1,-2)$ is a point of inflection.

End behaviour. As $x\to\infty$ , $f(x)\to\infty$ ; as $x\to-\infty$ , $f(x)\to-\infty$ (cubic with positive leading coefficient).

The sketch rises to the maximum at $(0,0)$ , falls through the inflection at $(1,-2)$ to the minimum at $(2,-4)$ , then rises through $(3,0)$ . Markers reward all labelled features and a curve whose shape matches the concavity.

Asymptotes for non-polynomial curves

Polynomials have no asymptotes, but rational and exponential functions do. For a rational function, a vertical asymptote occurs where the denominator is zero (and the numerator is not), and the horizontal asymptote is found from the behaviour as $x\to\pm\infty$ . For $y=e^{x}$ the line $y=0$ is a horizontal asymptote as $x\to-\infty$ .