AHSGE+Math+notes

**Order of Operations** works by first solving parenthesis, exponents, division, multiplication, addition, and then subtraction in that order Ex. 6 + (12 – 8)2 – 16 / 4 = 6 + (4)2 – 16 / 4 = 6 + 16 – 16 / 4 = 6 + 16 – 4 = 18 **Basic rules with negative numbers are as follows**: A negative times a negative equals a positive. A negative times a positive equals a negative. A positive times a positive equals a positive. Ex. –3 x –4 = 12 and –5 x 6 = -30 and 4 x 4 = 16 -23 + -4 = -27 and -20 + 6 = -14 and 85 + -60 = 15 and 4 - -1 = 5 **Mean** is a term that stands for average Ex. The mean of 34, 45, 27, and 36 is 35.5 (34 + 45 + 27 + 36) / 4 = 35.5 **Mode** is the numbers that occurs the most in a situation Ex. Bob scored 75, 77, 83, 75, 44, 90, and 75 on his tests. The mode of the tests is 75 (occurs more than the other grades) **Median** is the number in the middle of a group of numbers Ex. Rita scored 34, 44, 87, 56, 75, 66, and 54 on her tests. The median score is  56  (put in order from lowest to highest and pick the middle) **Greatest Common Factor** is the expression of highest degree and the largest integer coefficient that is a factor of each term Ex. 6//x//4 + 3//x//3 – 21//x//2 has a GCF of 3//x// **Monomial** is term that has no variable in its denominator and has only whole number exponents on its variables Ex. 5//x//, -7//y//2, 4, 1/8 //x//2//y//2 ** Polynomial ** is any monomial or algebraic sum of monomials Ex. 2 + 3//x//2, 5//x//2//y// + 4//xy//2 and 3//x// – 4//w//2 + 5//xy// **Linear Equations** are first-degree equations in two variables and their graphs will be straight lines = Ex. 2//x// + 3//y// = 4, //x// = -1, and //y// = 3//x// + 5 = = The Standard Form of a Linear Equation is A//x +// B//y =// C = The Linear Function equation is in the form of //y =// m//x +// b (//m// is the slope RISE     over RUN) ** Relation ** is a set of ordered pairs Ex. (1,3), (2,2), or (-4, 3) ** Domain ** of a relation is the set of all first components in the relation (the //x// value) Ex. In the above example, the domain is 1, 2, -4 In graphing, the x-axis is the domain axis ** Range ** of a relation is the set of all second components in the relation (the //y// value) Ex. In the Example for relation the range is 3, 2, 3 In graphing, the y-axis is the range axis ** Functions ** are relations in which each domain element has only one corresponding range element. It is a relation in which each first component appears only once. Also a relation in which no two ordered pairs have the same first component Ex. //r// = {(2, 3), (1, 6), (2, 2.5), (0, -1)} is not a function because the component, 2, appears more than once. If there is more than one x coordinate that is the same then there is no function //s// = {(0,0), (1,1), (2,4), (3,9)} is a function because each first component has only one corresponding second component Can tell if a graph is a function by the **vertical line test**. If any vertical line intersects a graph of a relation in more than one point, then the relation graphed is     not a function //f//(//x//) = 3//x// + 5 is the same as //y// = 3//x// + 5 (//y// and //f//(//x//) is the same) ** Area ** measures the space occupied by a two-dimensional region Ex. Area of rectangle is equal to length times width (A = l x w)     Area of a triangle is ½ height times base (1/2 h x b)      Area of a trapezoid is ½ height times the sum of the top and bottom lengths Area of a circle is equal to pi times the radius squared (pi = 3.14) ** Perimeter ** of a region is the length of its boundary. Sum of the length of the sides of a polygon = Ex. The perimeter of a square is 4 times the length of the side = **Circumference** of a circle is the entire length around the circle Ex. Circumference is equal to pi times diameter (across circle at the center) Circumference is equal to pi times two times radius (halfway across circle) ** Surface Area ** is the amount of material on the outside of a container Ex. SA of a rectangular prism is the perimeter times height plus two times area of      the base (S.A. = Ph + 2B) SA of a cylinder is two times pi times the radius times the height plus two times pi times radius squared (S.A. = 2~ rh = 2~ r2) = Volume measure how much an object will hold = Ex. Volume of a rectangular prism is equal to length times width times height (V = l x w x h) =   Volume of a cylinder is equal to pi times radius squared times height = (V = ~ r2h)
 * __ Math Basics  __**

Angles
Sum of angles in a triangle are equal to 180 degrees Supplementary angles are equal to 180 degrees (a + b = 180) Complementary angles are equal to 90 degrees (a + b = 90) If the lines q and r are parallel q   a  b a + c = 180 c d a = d and b = c     r   ** Distance ** is equal to the rate times the time to go the distance Ex. If a car travels at 55 miles per hour for 2.5 hrs, it will have traveled a distance of 137.5 miles. d = 55 mi/hr x 2.5 hr  ** Pythagorean Theorem ** enables you to find the length of the hypotenuse of a right triangle if you know the lengths of its legs  c    Ex. a2 + b2 = c2 a       b   ** Sum of Measures of Interior Angles of a Convex Polygon ** can be found by using a formula that uses the number of sides of the polygon Ex. S = 180 (n - 2) S = sum of interior angles of the polygon n = number of sides of the polygon 180 (8 – 2) = 1080 degrees ** FOIL Method ** is a way of factoring a trinomial by trial-and-error = Ex. x2 + 17x + 30  factored into (x + 15) (x + 2) = =  x2 + 11x + 30   factored into (x + 6) (x + 5) = **Distance Formula** is used to find the distance between two points (x1, y1) and (x2, y2) in the coordinate plane

=  Ex. D = (x2 –x1)2 + (y2 – y1)2  = ** Slope ** of a line is the tilt of a non-vertical line = Ex. the slope of the line through (x1, y1) and (x2, y2), with x1 = x2 = (y2 – y1) / (x2 – x1) = m m = slope ** Midpoint Formula ** can be found if a segment of line has endpoints (x1, y1) and (x2, y2) Ex. (x1 + x2) (y1 + y2) M = midpoint M = , 2  2

Forms of Equations
Ex. Standard form of an equation of a line: Ax + By = C     Slope-intercept form of an equation of a line: y = mx + b      Point-slope form of an equation of a line: y – y1 = m(x – x1) ** Quadratic Equation ** //a//x2 + //b//x + //c// = 0 can be solved for x in terms of //a//, //b//, and //c// by completing the square. The result is called the quadratic formula -b + b2 – 4ac Ex. x = 2a