Elementary Algebra

A linear equation is an equation where the variable appears only to the first power. Its general form is a times x plus b equals zero, where a and b are constants. To solve, isolate the variable by performing inverse operations on both sides: first subtract b from both sides, then divide both sides by a. Graphically, the equation y equals m times x plus b represents a straight line. The parameter m is the slope — it measures steepness as rise over run. The parameter b is the y-intercept — the point where the line crosses the vertical axis. A positive slope rises left to right. A negative slope falls. A zero slope is horizontal.

A quadratic equation has the form a times x squared plus b times x plus c equals zero. Its graph is a U-shaped curve called a parabola. If the leading coefficient a is positive, the parabola opens upward like a smile. If a is negative, it opens downward like a frown. The turning point is called the vertex, located at x equals negative b divided by two a. The discriminant is the expression under the square root in the quadratic formula: b squared minus four a c. If the discriminant is positive, there are two distinct real solutions (the parabola crosses the x-axis twice). If zero, there is exactly one real solution (the vertex touches the x-axis). If negative, the solutions are complex numbers (the parabola never touches the x-axis).

A function is a rule that assigns exactly one output to each input. We write f of x to mean "the value of the function f at input x." The input x is the independent variable — you choose it freely. The output is the dependent variable — its value depends on x. The domain is the set of all allowed inputs. The range is the set of all possible outputs. Functions can be represented four ways: as an equation, as a graph, as a table of values, or as a verbal description. Each representation shows a different perspective on the same relationship. The vertical line test determines if a graph represents a function: any vertical line must intersect the graph at most once.

A system of equations is a set of two or more equations that share the same variables. The solution is the ordered pair where all equations are true simultaneously. Graphically, this is the intersection point of the lines. Three algebraic methods exist for solving: substitution (solve one equation for a variable and plug into the other), elimination (add or subtract equations to cancel one variable), and graphing (draw both lines and find where they cross). A system can have one solution (intersecting lines), no solution (parallel lines that never meet), or infinitely many solutions (the same line written two ways).

Watch how the shape of a graph changes as you vary parameters. Adjust the coefficient to see stretching, compressing, and reflecting.