Patterns, Algebra, and Functions
What to know
Expect linear and quadratic relationships, slope and intercepts, systems of equations, inequalities, and function notation. You should also handle arithmetic and geometric sequences, direct and inverse variation, and translating word problems into equations. Understanding why a procedure works beats memorizing it - many items test whether you can explain or diagnose a student's error.
Traps to watch
- Slope sign and formula: $m = \frac{y_2-y_1}{x_2-x_1}$; keep the subtraction order consistent in numerator and denominator.
- Inequalities flip when you multiply or divide by a negative.
- Function notation: $f(x+2)$ means substitute $(x+2)$ for every $x$, not add 2 to the output.
- Systems: parallel lines (same slope, different intercept) → no solution; identical lines → infinitely many.
- Sequences: arithmetic adds a constant difference; geometric multiplies by a constant ratio. Don't assume a pattern is linear from two terms.
- Extraneous-looking answers: always check solutions in the original equation, especially after squaring.
Worked example
Line through $(2, 5)$ and $(6, 13)$: slope $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$. Using point-slope: $y - 5 = 2(x - 2)$, so $y = 2x + 1$. Check: at $x=6$, $y = 13$. ✓ Rule of thumb: after deriving an equation, plug both original points back in - it takes seconds and catches sign slips.