Patterns, Algebraic Relationships, and Functions
Function families and representations
Candidates must connect symbolic, graphical, tabular, and verbal forms of linear, quadratic, polynomial, rational, radical, exponential, logarithmic, and piecewise functions. Know domain and range restrictions, intercepts, slope and average rate of change, end behavior, asymptotes, transformations, composition, and inverses. Solve equations, inequalities, systems, and real-world models, and interpret parameters in context. Recognize arithmetic and geometric sequences and connect them to linear and exponential growth.
Traps
- f(g(x)) is not generally g(f(x)); compose in the stated order.
- An inverse exists only on a one-to-one domain; restrict domains deliberately, such as x ≥ 0 for squaring.
- Exponential growth eventually dominates any polynomial; do not extrapolate linearly.
- Squaring or clearing denominators can create extraneous solutions.
- In vertex form y = a(x − h)² + k, the vertex is (h, k), not (−h, k).
Worked example
For y = 2x² − 8x + 5, complete the square: y = 2(x² − 4x) + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3. The vertex is (2, −3), the axis is x = 2, and the minimum value is −3 because a > 0. Rule of thumb: the axis of symmetry is the average of the x-intercepts.