Functions | HSC Extension 1

Assessment evidence

What these examples cover

These examples show assessable skills, not an official HSC topic weighting or prediction. The 2024 NSW syllabuses have their first HSC examination in 2027; students sitting the 2026 HSC continue under the applicable 2017 syllabus.

Teacher check

Teacher verification note: check the mathematics, notation, permitted technology and syllabus fit for your cohort before classroom or assessment use.

3 deterministic examples

Functions questions with worked solutions

Each question is generated from an existing curriq QDSL archetype at a fixed seed. Marks and catalog content mappings travel with the generated question.

Worked example 1

1 markext1-f2-2

Third zero A cubic has leading coefficient 2, a repeated zero at

x=2

, and sum of zeros 3/2.

Part a · 1 mark

Find the third zero.

Worked solution

Answer: -5/2

Vieta: -5/2

Catalog reference: ext1_sgs25_third_polynomial_zero · seed 41001

Worked example 2

2 marksext1-f2-2

Extracting a quadratic factor A cubic

x^3-2x^2-2x+1

has root

x=-1

. Find the quadratic equation satisfied by its other two roots.

Part a · 2 marks

Factor out $(x+1)$ and state the remaining quadratic equation.

Worked solution

Answer: x^2 - (3 x) + 1

Factor Theorem: Factor out $(x+1)$ .
Polynomial Factorisation: $x^2-3x+1=0$ .

Catalog reference: ext1_ind_2025_q12bii_factor_cubic · seed 41001

Worked example 3

3 marksext1-f1-2

Nested absolute values Solve

||x|-5|<=3

. The complete solution is

[-8,-2] union [2,8]

once the two positive endpoints are established.

Part a · 1 mark

Find the positive inner endpoint.

Worked solution

Answer: 2

Absolute Value Inequality: $-3<=|x|-5<=3$ .

Part b · 2 marks

Find the positive outer endpoint and hence state the symmetric union of intervals.

Worked solution

Answer: 8

Compound Inequality: $2<=|x|<=8$ .
Absolute Value Solution Set: $[-8,-2] union [2,8]$ .

Catalog reference: ext1_ind_2025_q11d_nested_absolute_inequality · seed 41001

Content mapping and common errors

Functions content identifiers

ME-F1

Further Work with Functions

Work with inequalities; apply absolute value to equations and inequalities; sketch curves from simple relations.

Common student mistakes

·Multiplying both sides of an inequality by a negative without flipping the sign
·Incorrect absolute value inequality solution set

ME-F2

Polynomials

Apply the factor and remainder theorems; find roots; understand relationships between roots and coefficients.

Common student mistakes

·Incorrect sign when applying the remainder theorem
·Confusing sum of roots (α+β = −b/a) with product of roots (αβ = c/a)

Source and methodology

Examples are deterministic curriq catalog generations from existing QDSL archetypes at the published fixed seed. The generator copies only student-facing prompts, mark allocations, topic mappings, display tables or options, checked answers and marking-path results into this resource.

Review the official NSW Curriculum source ↗

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Question design

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Practice

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What these examples cover

Functions questions with worked solutions

Worked example 1

Worked example 2

Worked example 3

Functions content identifiers

Further Work with Functions

Polynomials

Source and methodology

Other Extension 1 topics

From the blog

Generate a Functions worksheet in 30 seconds