Patterns, Algebraic Relationships, and Functions
The Patterns, Algebraic Relationships, and Functions domain tests whether you can move fluently among verbal descriptions, tables, graphs, and symbolic rules, and whether you can solve and interpret equations, inequalities, and linear relationships in context.
Patterns and sequences
- Arithmetic sequences change by a constant difference d; geometric sequences change by a constant ratio r.
- The nth term of an arithmetic sequence is a + (n − 1)d, where a is the first term. For a geometric sequence it is a·r^(n−1).
Expressions and equations
- Combine like terms before solving; the equals sign represents balance, so every operation must be applied to both sides.
- Undo addition and subtraction first, then multiplication and division, to isolate the variable.
Linear functions
- Slope is the constant rate of change: m = (y₂ − y₁)/(x₂ − x₁), with change in y in the numerator.
- Slope-intercept form y = mx + b gives slope m and y-intercept b.
- A proportional relationship has the form y = kx and graphs as a line through the origin; any line not through the origin is linear but not proportional.
Worked example
Solve 3(x − 2) + 4 = 2x + 9. Distribute: 3x − 6 + 4 = 2x + 9, so 3x − 2 = 2x + 9. Subtract 2x from both sides: x − 2 = 9. Add 2: x = 11. Check by substitution: left side 3(9) + 4 = 31; right side 2(11) + 9 = 31. Both sides agree, so the solution is correct.
Traps that cost marks
- Reversing slope by putting the x-change over the y-change.
- Assuming a pattern is linear from only two or three data points.
- Forgetting to flip an inequality symbol when multiplying or dividing both sides by a negative number.
- Confusing the y-intercept with the slope when reading a real-world graph.