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
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide