A function table shows inputs 1, 2, 3, 4 and outputs 5, 8, 11, 14. A student checks only the first row and concludes the rule is 'output = input + 4.' What is wrong with this approach?
AThe rule should always involve multiplication rather than addition
BThe student should have checked the last row instead of the first
CThe rule works for (1, 5) but fails for (2, 8): 2 + 4 = 6, not 8. A rule must be verified against every row in the table
DAddition rules cannot produce outputs greater than 10
Checking only one pair is the most common error in finding function rules. The input-output pair (1, 5) satisfies many rules — output = input + 4, output = 5, output = 5 × input, output = input² + 4, etc. The discipline of verifying against all rows eliminates coincidental matches. The correct rule here is y = 3x + 2: the constant difference between consecutive outputs is 3 (linear), and checking (1,5): 3(1) + 2 = 5 ✓, (2,8): 3(2) + 2 = 8 ✓. Always check every row.
Question 2 Multiple Choice
A table shows inputs 1, 2, 3, 4 with outputs 3, 9, 27, 81. How can you tell this is NOT a linear rule, and what is the correct rule?
AThe rule is y = 3x because outputs are always multiples of 3; linear rules have outputs that are multiples of the multiplier
BThe rule is y = 3^x because consecutive outputs have a constant ratio of 3 (not a constant difference), which indicates exponential growth
CThe rule is y = x³ because the outputs 3, 9, 27, 81 are cubes of 3
DThe rule is y = x + 2 because the differences are constant at 6, 18, 54
Linear rules have a constant *difference* between consecutive outputs (e.g., +3, +3, +3...). Exponential rules have a constant *ratio* (e.g., ×3, ×3, ×3...). Here: 9/3 = 3, 27/9 = 3, 81/27 = 3 — constant ratio of 3, so the rule is y = 3^x. The misconception in option A is confusing 'multiples of 3' with linear growth; y = 3x would give 3, 6, 9, 12 (constant difference of 3), not 3, 9, 27, 81.
Question 3 True / False
Each row in a function table corresponds to a coordinate pair (input, output) = (x, y), and plotting all such pairs produces the graph of the rule.
TTrue
FFalse
Answer: True
This connection is direct and important: function tables and graphs are two representations of the same relationship. Each table row is one point on the graph. Linear rules (constant difference) plot as straight lines; exponential rules plot as curves; quadratic rules plot as parabolas. Understanding this equivalence means that learning to read tables now directly builds the intuition needed for graphing equations in the next course.
Question 4 True / False
If a rule correctly predicts the output for the first input-output pair in a table, it is the correct rule for the entire table.
TTrue
FFalse
Answer: False
Many different rules can produce the correct output for a single input. For example, given (2, 8), the rule could be y = 4x, y = x³, y = x + 6, y = 2x + 4, or infinitely many others. The correct rule must work for every row. This is not just a procedural check — it reflects the underlying concept: a function rule is a relationship that holds universally for all inputs, not a coincidence that happens to work once.
Question 5 Short Answer
How would you find the rule for a linear function table, and why must you verify the rule against every row rather than stopping after the first match?
Think about your answer, then reveal below.
Model answer: To find a linear rule: (1) Calculate the difference between consecutive outputs. If the differences are constant, the multiplier (slope) equals that constant difference. (2) Use one input-output pair to find the constant: multiply the input by the slope and compare to the output; the difference is the constant term. This gives the rule y = mx + b. You must verify against every row because any single pair is consistent with infinitely many rules. The correct rule is the one that describes the universal pattern — a relationship that holds for all inputs — not just a coincidence that works for one. A rule that fails any row is not the rule.
The two-step algorithm (constant difference → slope, then one pair → constant) is the systematic method for linear rules. The verification requirement is an instance of a broader mathematical discipline: a proposed rule is only valid when it passes all tests, not just selected ones. This same standard applies throughout algebra — a solution to an equation must satisfy the equation, not just look plausible.