Algebra and Functions
Algebra and Functions
This domain tests how well you translate situations into symbols, solve and interpret equations and inequalities, and read functions in symbolic, tabular, and graphical form. Expect linear relationships, systems, quadratic behavior, exponent patterns, and function notation, always with an eye on what the answer means in context.
Core rules to hold
- Equivalent expressions: combine like terms, apply the distributive property in both directions, and factor to reveal structure. Difference of squares: a^2 - b^2 = (a - b)(a + b). A trinomial x^2 + bx + c factors into (x + m)(x + n) when mn = c and m + n = b.
- Linear functions: slope = (y2 - y1)/(x2 - x1), the constant rate of change. Slope-intercept form is y = mx + b; point-slope form is y - y1 = m(x - x1). Parallel lines share m; perpendicular slopes are negative reciprocals.
- Inequalities: solving proceeds like equations except that multiplying or dividing both sides by a negative number reverses the inequality sign. Graph solutions with open or closed endpoints matching the symbol.
- Systems: substitution or elimination; the solution is the intersection point. Parallel lines give no solution, identical lines give infinitely many.
- Functions and domain: a relation is a function if each input has exactly one output. Exclude values that make denominators zero or radicands negative. The notation f(3) means substitute 3 for x, not multiply.
- Quadratic features: zeros from factored form, vertex at x = -b/(2a), and the sign of a tells whether the parabola opens up or down.
Worked example
A line passes through (2, 5) and (6, 13). Slope = (13 - 5)/(6 - 2) = 8/4 = 2. Using point-slope form: y - 5 = 2(x - 2), so y = 2x + 1. Check with the other point: 2(6) + 1 = 13, which matches. The slope of any perpendicular line is -1/2, and the y-intercept is 1, so the graph crosses the y-axis at (0, 1).
Traps that cost marks
- Computing slope as change in x over change in y.
- Dropping a negative sign when distributing, for example writing 3 - (x - 2) as 3 - x - 2.
- Forgetting to reverse an inequality when dividing by a negative.
- Ignoring excluded domain values in rational expressions.
- Assuming a linear model when growth is multiplicative; test successive ratios as well as successive differences.