Patterns, Algebra, and Functions
Patterns, Algebra, and Functions
This domain tests pattern recognition, algebraic manipulation, equation solving, and function behavior. Questions may ask you to generalize a sequence, model a situation, interpret slope or intercepts, or compare linear, quadratic, and exponential growth.
Core rules and concepts
- Arithmetic sequences have a constant difference d: a_n = a_1 + (n - 1)d. Geometric sequences have a constant ratio r: a_n = a_1 r^(n - 1).
- Linear functions have constant slope m = (y_2 - y_1)/(x_2 - x_1); forms include slope-intercept y = mx + b and point-slope y - y_1 = m(x - x_1).
- Quadratic functions have a vertex, axis of symmetry, and discriminant b² - 4ac that determines the number of real roots.
- Exponential functions grow or decay by a constant factor; they are not linear even when tables look regular.
- Domain restrictions matter: denominators cannot be zero, and even roots require nonnegative radicands.
- Systems of equations can be solved by substitution, elimination, or graphing; solutions must satisfy every equation.
Worked example
A linear function passes through (2, 5) and (6, 13). Find x when y = 19.
Slope: m = (13 - 5)/(6 - 2) = 8/4 = 2. Using point-slope: y - 5 = 2(x - 2), so y = 2x + 1. Set 19 = 2x + 1; 18 = 2x; x = 9. Check: (9, 19) lies on the same line because the slope from (6, 13) to (9, 19) is 6/3 = 2.
Traps that cost marks
- Reversing slope as rise/run incorrectly or subtracting coordinates in inconsistent order.
- Distributing a negative incorrectly, e.g., -(x - 3) = -x + 3.
- Confusing f(x + h) with f(x) + h.
- Forgetting to check for extraneous solutions after squaring both sides.
- Assuming exponential data are linear because successive y-values increase by a constant amount.