Linear functions across representations
Recover a constant-rate function from data and interpret both its input and output.
What you will tackle
- Create linear functions and interpret slope, intercept and input/output pairs
- Move between tables, graphs, rules and contextual descriptions
- Find inputs from outputs and outputs from inputs
- Derive a rule from two pairs or one pair and a rate
A function assigns one output to each input
In f(t) = 80 − 6t, t could be minutes and f(t) the litres remaining. The intercept is the initial 80 L and the slope is −6 L/min. Solving f(t) = 50 finds an input; evaluating f(5) finds an output. A graph, table and equation should encode the same pairings.
Use differences to recover the rule
For a linear function, equal input changes give equal output changes. A table with pairs (2,11), (5,23), (8,35) has slope 12/3 = 4. Substituting one pair into y = 4x + b gives b = 3. A contextual domain may stop the line before negative time or volume occurs.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
A table contains (2,11), (5,23) and (8,35). Find f(x), f(10), and the x with f(x) = 47.
Show a starting hint
A table of a line has a constant rate, not necessarily equal raw differences.
Compare with the explained answer
- The constant slope is 4; using (2,11) gives intercept 3.
- f(10) = 4(10) + 3 = 43.
- 4x + 3 = 47 gives x = 11.
Answer / response criteria: f(x) = 4x + 3; 43; x = 11.
Independent practice
Independent 1
g(3) = 19 and g has slope −2. Find its rule, intercept and g(0).
Compare with the explained answer
- Write g(x) = −2x + b.
- 19 = −6 + b gives b = 25.
- The vertical intercept and g(0) are both 25.
Answer / response criteria: g(x) = −2x + 25; intercept (0,25); g(0) = 25.
Application practice
Application 1
A tank contains 80 L and loses 6 L/min. How long until 50 L remains? State a physical domain and describe the graph.
Compare with the explained answer
- Use V(t) = 80 − 6t and solve 80 − 6t = 50.
- 6t = 30, so t = 5 min.
- The graph decreases from (0,80) to (40/3,0); the physical domain is 0 ≤ t ≤ 40/3.
Answer / response criteria: 5 min; 0 ≤ t ≤ 40/3; a descending line segment.
What can you explain now?
A tank contains 80 L and loses 6 L/min. How long until 50 L remains? State a physical domain and describe the graph.
Compare with the explanation
5 min; 0 ≤ t ≤ 40/3; a descending line segment.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide