Combining graphical/algebraic/calculus information in sketches | Year 12

01 / LEARN

The idea, step by step

Plan a sketch instead of guessing

State the domain and exact key points before drawing. Add asymptotes as dashed guides and label turning points, holes and intercepts. A sign or derivative table can organise the branches without relying on dozens of plotted points.

Use a calculator graph as a check, not as a substitute for exact features. A window can hide a hole, compress an extremum or make a branch look continuous across a pole.

Check the final picture

Compare the drawn graph with the function sign, derivative sign, symmetry and end behaviour. A tangent describes only a neighbourhood of a point, not the entire curve.

State when an apparent feature is missing: there may be no intercept, no real stationary point or no vertical asymptote. Avoid forcing every graph into the same familiar shape.

A common trap

Letting a calculator connect separate branches across an excluded input.

Instead: Identify the domain and asymptotes algebraically before drawing.

02 / WORKED EXAMPLES

See the thinking

Example 1

A complete reciprocal-plus-line sketch

Sketch $y = x + 1 / x$ using its domain, extrema and asymptotes.

The equation is undefined at 0.
The difference from y=x is 1/x, tending to zero at either end.
Derivative signs determine each branch shape.

Two odd-symmetric branches; x=0 is vertical, y=x is oblique; extrema

(- 1, - 2)

and

(1, 2)

; no intercepts.

Example 2

A bounded positive curve

For $y = 1 / (x^{2} + 1)$ , justify the complete graph from symmetry, derivative and limits.

$y^{'} = - 2 x / {(x^{2} + 1)}^{2}$ .
The derivative is positive for negative x and negative for positive x.
The denominator never vanishes.

An even positive curve, maximum

(0, 1)

, approaching y=0 at both ends with no vertical asymptote.

Check your understanding

Does a horizontal asymptote describe a local tangent?

Show answer

No.

It describes end behaviour as input magnitude grows.

03 / WORKSHEET

Your turn

6 questions · 23 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Sketch $y = x + 1 / x$ using its domain, extrema and asymptotes.
5 marks
Give a complete sketch description of $y = ∣ x - 2 ∣ - 3$ .
4 marks
A student draws one continuous curve through x=1 for $y = x^{2} / (x - 1)$ . Explain and correct the sketch.
3 marks
For $y = 1 / (x^{2} + 1)$ , justify the complete graph from symmetry, derivative and limits.
4 marks
Two formulas simplify to the same expression. Explain why their graphs might still differ.
3 marks
Find the tangent to $f (x) = x / (x^{2} + 1)$ at x=0 and explain why it does not determine the whole graph.
4 marks

Read the worked solutions

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

Sketch $y = x + 1 / x$ using its domain, extrema and asymptotes.
5 marks
The equation is undefined at 0.
The difference from y=x is 1/x, tending to zero at either end.
Derivative signs determine each branch shape.
Two odd-symmetric branches; x=0 is vertical, y=x is oblique; extrema $(- 1, - 2)$ and $(1, 2)$ ; no intercepts.
Give a complete sketch description of $y = ∣ x - 2 ∣ - 3$ .
4 marks
Use the two linear branches and place the exact intercepts.
The function is continuous but not differentiable at x=2.
V shape with vertex $(2, - 3)$ , roots -1 and 5, y-intercept -1; slopes -1 then 1; range $[- 3, \infty)$ .
A student draws one continuous curve through x=1 for $y = x^{2} / (x - 1)$ . Explain and correct the sketch.
3 marks
The original denominator is zero at 1.
The numerator is nonzero there, giving a pole rather than a hole.
The graph must be broken into branches separated by the vertical asymptote x=1.
For $y = 1 / (x^{2} + 1)$ , justify the complete graph from symmetry, derivative and limits.
4 marks
$y^{'} = - 2 x / {(x^{2} + 1)}^{2}$ .
The derivative is positive for negative x and negative for positive x.
The denominator never vanishes.
An even positive curve, maximum $(0, 1)$ , approaching y=0 at both ends with no vertical asymptote.
Two formulas simplify to the same expression. Explain why their graphs might still differ.
3 marks
Compare $x + 1$ with $(x^{2} - 1) / (x - 1)$ .
Only the second excludes x=1.
Their original domains may differ, producing a hole in one graph.
Find the tangent to $f (x) = x / (x^{2} + 1)$ at x=0 and explain why it does not determine the whole graph.
4 marks
$f (0) = 0$ and $f^{'} (0) = 1$ .
A tangent is local information; extrema and end behaviour need separate analysis.
$y = x$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist functions and sketching graphs. Calculus-based analysis assumes concurrent Methods differentiation; real-polynomial division detail is taught in the separate Real Polynomials chapter.

Official curriculum source ↗

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