Skip to main content
Back

College Algebra Homework Key Concepts Cheat Sheet

Control buttons has been changed to "navigation" mode.
1/17

Terms in this set (17)

  • Graphs & Utilities

    Graphs represent equations visually. Utilities include tools like graphing calculators to analyze functions and equations.

  • Linear Equations & Rational Equations

    Linear equations have the form \(ax + b = 0\). Rational equations involve ratios of polynomials.

  • Models & Applications

    Use algebraic models to represent real-world problems and solve applications using equations and functions.

  • Complex Numbers

    Complex numbers are numbers in the form \(a + bi\), where i is the imaginary unit with \(i^2 = -1\).

  • Quadratic Equations

    Quadratic equations have the form \(ax^2 + bx + c = 0\). Solutions can be found by factoring, completing the square, or the quadratic formula.

  • Other Types of Equations

    Includes polynomial, radical, exponential, and logarithmic equations, each requiring specific solving techniques.

  • Linear Inequalities & Absolute Value Inequalities

    Linear inequalities use <, >, ≤, ≥ to compare expressions. Absolute value inequalities involve expressions like \(|x| < a\) or \(|x| > a\).

  • Basics of Functions & Their Graphs

    A function assigns exactly one output to each input. Graphs show the relationship between input (x) and output (f(x)).

  • More on Functions and Their Graphs

    Includes domain, range, and behavior of functions such as increasing, decreasing, and symmetry.

  • Linear Equations & Slope

    Slope measures the steepness of a line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Linear equations can be written in slope-intercept form \(y = mx + b\).

  • More on Slope

    Includes parallel lines (equal slopes) and perpendicular lines (slopes are negative reciprocals).

  • Transformations of Functions

    Function graphs can be shifted, reflected, stretched, or compressed by modifying the function's equation.

  • Combinations of Functions; Composite Functions

    Combination involves adding, subtracting, multiplying, or dividing functions. Composite functions are formed by applying one function to the result of another: \((f \circ g)(x) = f(g(x))\).

  • Inverse Functions

    An inverse function reverses the effect of the original function, satisfying \(f(f^{-1}(x)) = x\).

  • Distance & Midpoint Formulas; Circles

    Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Midpoint formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Circle equation: \((x - h)^2 + (y - k)^2 = r^2\).

  • Systems of Linear Equations in Two Variables

    Solving two linear equations with two variables by substitution, elimination, or graphing to find the point of intersection.

  • Systems of Linear Equations in Three Variables

    Solving three linear equations with three variables using substitution or elimination to find the common solution.