Algebraic Properties Identities: 0 & 1 Inverses: Opposites & Reciprocals Subtraction Adding Opposites Division Multiplying Reciprocals Commutative Order of addition & multiplication Associative Multiple additions & multiplications Distributive a(b + c) = ab + ac
Simplify Expressions Remove grouping symbols. –Associative & Distributive Properties Combine like terms. –Distributive Property Complete all possible rational arithmetic.
Solve Linear Equations Simplify both sides of the equation. –Result: ax + b = cx + d Clear Fractions (Optional) Apply the addition principle & simplify –Move all linear terms to one side and all other terms to the other side. –To move a term to the other side, change its sign. –Result: ex = f Apply the multiplication principle & simplify –Divide both sides by the coefficient of the linear term. Check
Formulas Apply –Given values for all but one variable, determine the value of the remaining variable. Solve for a specified variable in terms of the other variables. –Same procedure as linear equation w/ one variable. –Treat all other variables like constants.
Solve Inequalities Same as solving equations with two exceptions –When multiplying or dividing both sides of the inequality by a negative number, the direction of the inequality is reversed. –When switching sides of an inequality, the direction of the inequality is reversed. Graphing inequalities … (a(a x > a ]a]a x a [a[a x a )a)a x < a
Rectangular Coordinates AKA … –Cartesian Coordinates –The XY-plane Ordered pairs ( x, y ) Lines x-intercept: ( a, 0 ) y-intercept: ( 0, b ) Slope:
Equations of Lines Standard Form: Ax + By = C x-intercept: (C/A, 0) y-intercept: (0, C/B) Slope-Intercept Form: y = mx + b y-intercept: (0, b) Slope: m Point-Slope Form: y – k = m(x – h) Point: (h, k) y = k + m(x – h) Slope: m Horizontal Lines: y = b Vertical Lines: x = a
Graphing Lines Option 1: y = mx + b –Plot the y-intercept: (0, b) –Find a second point using the slope (rise/run) –Draw the line Option 2: Ax + By = C –Plot the x-intercept: (C/A, 0) –Plot the y-intercept: (0, C/B) –Draw the line Checking … find and verify another solution. A picture of ALL of the solutions of an equation.
Solve Systems of Equations Graphing –Graph both equations. –The point of intersection is the solution. Substitution –Solve one equation for one variable. –Substitute into the other equation and solve. –Use the result to determine the other variable. Elimination –Multiply to get coefficients of a variable to be opposites. –Add the equations and solve. –Repeat for the other variable. Check (in both equations)
Functions f(x) = expression in x y = expression in x Domain: Set of all values for x. Range: Set of all possible results. f(3) –Replace all occurrences of x with 3 –Simplify. Linear Function: f(x) = mx + b
Polynomials - Terminology Term –constant term –linear term –quadratic term –leading term –leading coefficient Coefficient Degree –of a term –of a polynomial Polynomial –monomial –binomial –trinomial Evaluating a … –polynomial –polynomial function
Polynomials - Operations Addition –Add like terms Subtraction –Find the opposite of the polynomial being subtracted –Add Multiplication –Multiply each term of the first by each term of the second –Combine like terms Division –By monomial … divide each term (properties of exponents) –By polynomial … long division Synthetic division: Only with division by x-a or x+a
Polynomials FOIL “similar” – O & I terms are like terms (a + b)(a – b) = a 2 – b 2 (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 Special Products