Essential Algebra Formulas and Graphing Rules

Algebraic Operations and Equations

Completing the Square

To complete the square, take 1/2 of b, then square it. When a ≥ 1, remember to factor out the a first.

Exponents and Radicals

• Negative Exponents: Take the reciprocal of the base and make the exponent positive. In fractions, move the negative exponent to the bottom and make it positive.
• Radicals: Always check your answers.
• Dividing Radicals: Divide the numbers inside by the other inside, and divide the outside numbers by the other outside numbers.
• Radical Exponents: Only square or cube the number/variable under the radical unless there are parentheses.
• Conversion: To convert a radical into an exponent:
• Rationalizing: To get a radical off the bottom, multiply by the radical over itself (so it equals 1).

Inequalities and Notation

• Inequalities: When multiplying or dividing both sides by a negative number, flip the signs.
• Interval Notation: This notation uses infinity symbols and should be in the (-∞, x) ∪ (y, ∞) form. Use a bracket instead of a parenthesis if there is a filled-in dot (≥ or ≤).
• Compound Inequalities: Solve for x and graph the line on a number line.

The Discriminant

Formula: b² - 4ac

• If greater than 0: 2 answers
• If exactly 0: 1 answer
• If below 0: No real answers

Functions and Graphing

Functions: When evaluating, leave the function notation alone and solve the other side of the equation.

Systems of Equations

Elimination Method: Multiply or divide one equation to get a negative and positive of the same number with the same exponent, then combine and cancel. Cancel x's to find y, and cancel y's to find x. The answer should be a coordinate.

Graphing Linear Equations

• Ax + By = C: Plug in 0 for x to find the y-intercept, and plug in 0 for y to find the x-intercept.
• Horizontal and Vertical Lines: Horizontal lines are y = a, while vertical lines are x = a (with no slope).
• Point-Slope Form: Plug the slope into the formula y = mx + b, plug in points for x and y (where b remains as b), and solve for b. Then plug b into the original formula to get the equation of the line. You can also use:

Quadratic Functions

• Roots of a Parabola: Factor from standard form and solve for the x values.
• Parabola Direction: A positive a value means the parabola opens upwards.
• Graphing: Find the Axis of Symmetry (AOS) using -b/2a, then plug that and other numbers into an input/output chart.
• Vertex Form:

Piecewise Functions

• Graphing: Draw dotted lines and graph the lines on their respective sides. Fill in or leave the dot open depending on whether it is ≥/≤ or >/<.
• Evaluating: Look at f(x) and decide which category x belongs in, then plug it into the function.
• Graphical Evaluation: Look at the function (e.g., f(-2)) and find the corresponding value on the y-axis; note whether the dot is filled or open.

Sequences and Interest Formulas

Reminder: When using percentages for rates in sequences, turn them into decimals (which should be very small and below 1).

• Geometric Sequence Formula:
• Arithmetic Sequence Formula: (where d is the amount added or subtracted).
• Compound Continuously: (P = starting amount, r = rate, t = time).
• Compounded Periodically: (n is the number of times compounded per year, e.g., monthly = 12, quarterly = 4).
• Compounded Annually: