7.5 Patterns, Sequences & Functions

7.5 Patterns, Sequences & Functions

Quick Answer: A function pairs each input with exactly one output. Repeating an output is fine; repeating an input with two different outputs is not. On a graph, the vertical line test decides it. A recursive rule says how to get the next term from the one before; an explicit rule computes any term directly from its position, which is what lets you answer "what is the 100th term?" without listing 99 of them. A function is linear when its rate of change is constant, and proportional only when it is linear and passes through the origin.

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Where This Sits in the Blueprint

Pattern reasoning is one of the longest-running threads in the B.E.S.T. Standards, and it terminates in the grade 8 functions benchmarks:

Grade 8's Linear Relationships, Data Analysis, and Functions is the largest reporting category on the grade 8 test at 25-31%, so function reasoning carries real weight.

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Patterns: From Rule to Sequence

A numerical pattern is a sequence generated by a rule. Given a starting value and a rule, students generate terms; given terms, they infer the rule.

Example. Start at 4; the rule is "multiply by 3, then subtract 2."

$4 \rightarrow 10 \rightarrow 28 \rightarrow 82 \rightarrow 244$

Check: $4 \times 3 - 2 = 10$; $10 \times 3 - 2 = 28$; $28 \times 3 - 2 = 82$. Apply the whole rule each time - applying only part of it is the standard error.

Recursive vs. Explicit Rules

For the arithmetic sequence 3, 8, 13, 18, 23, ... the common difference is 5. The explicit rule is $a_n = 5n - 2$: check $n = 1$ gives $3$, and $n = 4$ gives $18$. The 100th term is $5(100) - 2 = 498$ - one computation, not ninety-nine.

Building an explicit rule for an arithmetic sequence:

$a_n = (\text{common difference}) \times n + (\text{first term} - \text{common difference})$

For 7, 11, 15, 19: the difference is 4, so $a_n = 4n + (7 - 4) = 4n + 3$. Verify at $n = 3$: $4(3) + 3 = 15$. Correct.

Two Patterns Side by Side

Grade 5 asks students to generate two patterns from two rules, form ordered pairs from corresponding terms, and plot them.

Every B-value is exactly 3 times its A-value. Plotted, the points fall on a straight line through the origin - the students' first encounter with a proportional relationship, years before it is named.

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Functions: The One-Output Rule

A function is a relation in which each input has exactly one output.

   FUNCTION                        NOT A FUNCTION
   input -> output                 input -> output
     2   ->   5                      2   ->   5
     3   ->   8                      3   ->   8
     4   ->   5   (repeat OK)        2   ->  -1   (input 2 has two outputs)
     6   ->  11

Repeated outputs are fine. Both $x = 3$ and $x = 7$ can map to $y = 4$. What breaks a function is one input with two outputs.

The Vertical Line Test

Sweep a vertical line across the graph. If it ever crosses the graph in more than one place, some input has two outputs and the relation is not a function.

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Linear vs. Nonlinear

A function is linear when its rate of change is constant. Three equivalent signals:

1. Equation: it can be written $y = mx + b$, with $x$ to the first power only.
2. Graph: a straight line.
3. Table: with evenly spaced inputs, the first differences in $y$ are constant.

The left column's differences are constant; the right column's are not. That is the whole test.

[!IMPORTANT] The inputs must be evenly spaced before you compare differences. A table listing $x = 1, 2, 5, 9$ will show unequal $y$-differences even for a perfectly linear function. Check the $x$ spacing first, every time.

Proportional Is a Subset of Linear

Every proportional relationship is linear. Not every linear relationship is proportional. The deciding question is whether the $y$-intercept is 0 - equivalently, whether $\frac{y}{x}$ is the same constant for every pair.

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Comparing Functions Across Representations

Grade 8 items regularly present two functions in different forms and ask which has the greater rate of change or the greater initial value. The strategy is to convert both to the same form - usually rate of change and initial value - before comparing.

Function A (equation): $y = 6x + 20$. Rate of change 6, initial value 20.

Function B (table):

Rate of change: $\frac{43 - 35}{2 - 0} = \frac{8}{2} = 4$. Initial value: 35, read directly at $x = 0$.

Comparison: A grows faster (6 per unit versus 4), but B starts higher (35 versus 20). B leads until A catches up. Setting $6x + 20 = 4x + 35$ gives $2x = 15$, so $x = 7.5$ - after which A is permanently ahead. A verbal description would be handled the same way: extract the rate and the starting value, then compare those two numbers.

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Common Exam Traps & Misconceptions

[!WARNING]

Trap 1: Thinking a repeated output breaks a function

The set ${(1, 4), (2, 4), (3, 4)}$ is a function. Three inputs share one output, which is allowed. Only a repeated input with different outputs disqualifies a relation.

[!WARNING]

Trap 2: Applying only part of a two-step rule

With "multiply by 3, then subtract 2," students often multiply for a few terms and then forget the subtraction. Write the rule at the top of the scratch paper and execute both steps every time.

[!WARNING]

Trap 3: Calling every straight-line relationship proportional

$y = 5x + 12$ is linear but not proportional, because it does not pass through the origin. Check the value at $x = 0$: if $y$ is anything other than 0, it is not proportional.

[!WARNING]

Trap 4: Comparing differences with unevenly spaced inputs

A table listing $x = 0, 1, 3, 6$ will show growing $y$-differences even for a perfectly linear function, because the input gaps are growing too. Divide the change in $y$ by the change in $x$ rather than eyeballing the $y$ column.

[!WARNING]

Trap 5: Confusing rate of change with initial value

In $y = 6x + 20$, the rate of change is 6 and the initial value is 20. An item asking which function "starts higher" wants the initial value; one asking which "grows faster" wants the rate. Read the question word before comparing.