Curve analysis | Year 12 | Ascent Library

01 / LEARN

The idea, step by step

Build a complete picture

Begin with the domain, intercepts and end behaviour. Find stationary points, increasing/decreasing intervals and concavity, then put the information together. A sketch should be consistent with all these features.

A technology graph is useful to check a conclusion, but the viewing window can hide roots, asymptotes or turning points. Justify the features analytically when asked.

Respect disconnected domains

A rational function may have separate branches. Include excluded inputs and asymptotes in a sign chart, even if no derivative is zero there.

Choose a helpful scale and label important coordinates clearly. Do not connect branches across a vertical asymptote or turn an open-domain boundary into a filled point.

A common trap

Joining graph branches through an excluded input.

Instead: A curve cannot pass through a value outside its domain, regardless of a plotting window.

02 / WORKED EXAMPLES

See the thinking

Example 1

A cubic sketch from its features

Describe a sketch of $y = x^{3} - 3 x$ .

The roots are $- \sqrt{3}, 0, \sqrt{3}$ .
The maximum is $(- 1, 2)$ and the minimum is $(1, - 2)$ .
There is an inflection at $(0, 0)$ ; the curve falls leftwards and rises rightwards.

An odd cubic with the three labelled roots and two stationary points.

Example 2

Separate branches of a rational curve

Find the key features of $y = x + 1 / x$ .

The domain excludes zero; asymptotes are $x = 0$ and $y = x$ .
The derivative $1 - 1 / x^{2}$ is zero at $x = \pm 1$ .
Sign changes give maximum $(- 1, - 2)$ and minimum $(1, 2)$ .

Two branches, with no real x-intercepts and the stated asymptotes.

Check your understanding

How many real x-intercepts has $y = x^{2} + 1$ ?

Show answer

None.

Its minimum is $(0, 1)$ , above the x-axis.

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find all intercepts of $y = x^{3} - 4 x$ .
2 marks
State increasing intervals for $y = x^{3} - 3 x$ .
2 marks
For $y = x^{4} - 2 x^{2}$ , find all stationary coordinates.
3 marks
Find the vertical and oblique asymptotes of $y = x + 1 / x$ .
2 marks
Sketch $y = e^{- x}$ by stating its intercept, monotonicity, concavity and asymptote.
4 marks
For $f (x) = x^{3} - 3 x$ on $[- 2, 2]$ , find the absolute maximum and minimum.
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find all intercepts of $y = x^{3} - 4 x$ .
2 marks
Factor as $x (x - 2) (x + 2)$ .
Set each factor to zero.
x-intercepts $(- 2, 0), (0, 0), (2, 0)$ ; y-intercept $(0, 0)$ .
State increasing intervals for $y = x^{3} - 3 x$ .
2 marks
The derivative is $3 (x^{2} - 1)$ .
It is positive when $∣ x ∣ > 1$ .
$(- \infty, - 1)$ and $(1, \infty)$ .
For $y = x^{4} - 2 x^{2}$ , find all stationary coordinates.
3 marks
The derivative is $4 x (x^{2} - 1)$ .
Evaluate the original function at $- 1, 0, 1$ .
$(- 1, - 1), (0, 0), (1, - 1)$ .
Find the vertical and oblique asymptotes of $y = x + 1 / x$ .
2 marks
The reciprocal term is undefined at zero.
It tends to zero as $∣ x ∣$ becomes large.
$x = 0$ and $y = x$ .
Sketch $y = e^{- x}$ by stating its intercept, monotonicity, concavity and asymptote.
4 marks
The first derivative is negative and the second positive.
The exponential remains positive and tends to zero as $x$ increases.
Intercept $(0, 1)$ ; decreasing; concave up; horizontal asymptote $y = 0$ .
For $f (x) = x^{3} - 3 x$ on $[- 2, 2]$ , find the absolute maximum and minimum.
4 marks
Check stationary inputs $- 1, 1$ and endpoints $- 2, 2$ .
The values are respectively $2, - 2, - 2, 2$ .
Maximum 2 at $x = - 1, 2$ ; minimum $- 2$ at $x = - 2, 1$ .

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Curriculum reference & lesson scope

SACE Stage 2 Mathematical Methods

Further differentiation and applications from the current SACE Methods outline. Logarithmic derivatives have their own chapter; integration follows later. Trigonometric calculus uses radians.

Official curriculum source ↗

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