Quantitative Reasoning Measure

Functions

An algebraic expression in one variable can be used to define a function of that variable. Functions are usually represented by letters like f, g, or h. ex: 3x+5 can be used to define a function f by f(x)=3x+5 where f(x) is the value of f at x and is obtained by substituting the value of x in the expression above. If x=1 is substituted then the result is f(1)=3(1)+5=8

*Domain* of a Function

Domain of a function is the set of all permissible inputs or all values of the variable x. Use this to see what numbers won't work in an equation for example: f(x)= 2x/x-6 Here x can't equal 6 because if you plug in 6 you would get 12/0 and that is not defined (can't do - no real number) SO the domain of f consists of all real #'s except for 6

Multiplying Two Algebraic Expressions

Each term is multiplied by each term of the second expression & the results are added. ex: (x +2)(3x-7) you do x(3x)+x(-7)+2(3x)+2(7) THEN simplify 3x^2+6x-14 Combined liked terms 3x^2-1x-14 so (x+2)(3x-7)=3x^2-x-14

Linear Equation

Equation involving one or more variables where each term in equation is either a constant or variable multiplied by a coefficient. *Can't be variables multiplied together (xy) or power raised greater than 1* Y=mx+b

Estimation & Percent

Estimation - You can estimate with stats but not with Geometry. Percent- Another type of fraction - percent is a fraction with a denominator of 100 so if you say 50% it is same as 50/100

Quadrilaterials

Every quadrilateral has four sides and four interior angles where the angles add up to *360 degrees*

Exponents

Exponents - want to multiply something by itself a number of times. (4 called base ^3 exponent). *Any exponent of 0 is 1* If bases are the same you can add the exponents.

Linear Equations in *One* Variable

Find simpler equation by combining like terms & applying rules until you solve solution. Can plug in answer to check Possible there is no solution Could look like a linear equation but actually an identity (true for all numbers when plugging them in)

Solving Quadratic Equations: By Factoring

Some can be done faster by factoring out. ex: 2x^2-x-6=0 can be factored as (2x+3)(x-2)=0 *When factoring think what times what is the c number & what subtracted/added by/to what is the b number* If you break up the equation at least one of the two pieces must equal zero.

Area of a Circle

Pie(r)^2

Part 2 *Algebra* Like terms

Have the same variables (letters) & corresponding variables have same exponents Can be combined by adding coefficients

Converting Decimals into Fractions

*For lower numbers/rounded numbers example:* 2.3 since each place value is a power of 10 you would do. . . 2.3= 2+ 3/10 (because the 3 is in the 10 place) THEN would give 2 a denominator of ten & multiply 2 by ten getting 20/10 + 3/10 = 23/10 ex #2: 90.17= 90 + 17/100 (because 17 is in the 100s place) THEN multiple 90 by 100 & give denominator of 100 9,000/100 + 17/100 = 9,017/100 ex #3: 0.612 = 612/1000

Part 1: *Arithmetic* Integers 1 x 1 = -1 x -1 = 1 x -1 =

(Not a decimal, a whole number.) - A positive # - A positive # - A negative # Ex: If r, s and t are integers and rs=t then r and s are factors, or divisors, of t also t is a multiple of r & s and t is divisible by r and s. The factors of an integer includive positive & negative integers.

Types of Integers: Prime Numbers

(greater than 1) Whole number with two factors, itself and 1. 2, 3, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. All even numbers greater than 2 are not prime numbers. If the last digit is even the number is divisible by 2.

Percent Decrease

*Amount Decrease/Base* If decrease 500 to 400 then 500-400/500 = 100/500=20/100=20%

Solving Linear Equations 3 Rules

*Find the values of the variables* 1.) When same constant added/subtracted from both sides of equation, new equation is equal to original equation 2.) Same as above but with multiplication/division 3.) When expression that occurs in an equation is replaced by an equivalent expression, the equality is equivalent to original equation. ex: 2(x+1) is same as 2x+2 and 2(x+1) can replace 2x+2 and its equal to the original equation

Solving Linear Inequalities: -Inequality -2 Rules of Solving Inequalities

*Inequality* - math statement that uses < less than, > greater than, less than or equal to, or greater than or equal to. To solve you simplify the inequality by isolating the variable on one side of the inequality by using two rules Rule 1: When same constant is added to or subtracted from both sides the direction of the inequality is preserved Rule 2: When same nonzero constant is multiplied/divided the direction is preserved is the constant is *positive* - direction is reversed if constant is *negative* (you only reverse if the last step you do the number is negative to begin with)

Coordinate Geometry: Origin & Quadrants

*Origin* - Point where two axes intersect (0) *Quadrants* - II, I, III, IV from left to right

Percent Change

*changes from positive amount to another positive amount* Can compute the amount of change as a % of initial amount

Simplifying Algebraic Expressions

- 1st Factor numerator & denominator - Cancel out things that occur twice ex: 7x^2+14x/2x+4 7x(x+2)/2(x+2) & cancel out x+2 since it occurs twice = 7x/2

For any Function h(x) & any positive # C the following are true:

- Graph h(x)+c is h(x) shifted upward by c units - Graph h(x)-c is h(x) shifted downward by c units - Graph h(x+c) is h(x) shifted to the left c units - Graph h(x-c) is h(x) shifted to the right c units f(x)=absolute value of x +2 would be y=absolute value of x sifted upwards by two units *Goes opposite direction on graph when in parathesis

Inscribed

A polygon is inscribed in a circle if all its vertices lie on the circle or the circle is circumscribed about the polygon. A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle, or the circle is inscribed in the polygon. If the center is on one of the sides that side is a diameter of the circle. If one side of an inscribed triangle is a diameter of the circle then the triangle is a right triangle. IF an inscribed triangle is a right triangle, then one of its sides is a diameter of the circle. Inscribed angel= 180 - (central angle/2)

Two Methods of Solving Linear Equations: 1.) Substitution

1.) 4x+3y=13 & x+2y=2 First Solve for Y x+2y = 2 -2y -2y x=2-2y *Substitute* 2-2y for x in first equation to get 4(2-2y) + 3y=13 *Simplify* *Combine like terms* to get 8-5y=13 *Solve* 8-5y= 13 -8 -8 -5y= 5 /5 /5 y=-1 *THEN* Use y=-1 to solve for x by substituting -1 for y in *second* equation to get x+2(-1)=2 x-2 = 2 +2 +2 x=4 x=4 & y=-1 OR (4,-1)

Types of Polygons: 1.) Triangle 2.) Quadrilateral 3.) Pentagon 4.) Regular Polygon

1.) Simplest polygon with 3 sides. Sum of the measures of the interior angles of a triangle is 180 degrees. 2.) Four sides & can be divided into 2 triangles by drawing a diagonal (so measures 360 degrees - because two triangles 180+180) 3.) 5 sides & can be divided into 3 triangles by selecting one of the vertices & drawing 2 line segments connecting the vertex to the two nonadjacent vertices. If a polygon has n sides it can be divided into n-2 triangles (540 degrees - because 3 trianlges 180 *3) 4.) Polygon in which all sides are congruent & all interior angles are congruent (octagon - 8 sides). So with an octagon you would do (8-2)(180)=1,080 THEN 1,080/8=135 degrees is the *measure of each angle*.

Rules of Exponents

1.) x^-a = 1/x^a 2.) (x^a)(x^b)=x^a+b 3.) x^a/x^b= x^a-b=1/x^b-a 4.) x^0=1 5.) (x^a)(y^a)= (xy)^a 6.) (x/y)^a= x^a/y^a 7.) (x^a)^b=x^a*b

Fractional Expressions

Pie/2 and Pie/3 to add them make the denominators the same ex: 6 is a common denominator so Pie(3)/2(3) = 3(pie)/6 & pie(2)/3(2)=2(pie)/6 SO 3(pie)/6 + 2(pie)/6 = 5(pie)/6

Triangles

3 sides & 3 interior angles. The length of each side must be less than the sum of the lengths of the other two sides (i.e. lengths could not be 4, 7, 12 because 4+7=11 so only 11 so 12 is greater than 4+7.

Mixed Numbers

4 3/8 which really means 4+3/8

Geometry: Acute Angle Obtuse Angle

Acute Angles - angle with measure less than 90 degrees Obtuse Angles - angle with measure between 90 degrees and 180 degrees

Factoring Out #'s or Variables

A # or a variable that is a factor of each term can be factored out ex: 4x + 12= 4(x + 3) OR 15y^2-9y= 3y(5y-3)

Arc of a Circle

Arc - Any two points on a circle containing the two points and all the points between them. Usually identified by at least three points Measure of an Arc - the measure of its central angle (center of the circle) which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc. *(arc length x 360/2PieR)* To find the Length of An Arc - The ratio of the length of an arc to the circumference is equal to the ratio of the degree measure of the arc to 360 degrees. Length of arc/circumference *(degree measure/360 x 2PieR)*

The Area of a Triangle

Area (A) of a triangle is given by the formula *A=(b)(h)/2* where b=lengths of a base h=lengths of the corresponding height Any side of the triangle can be used as a base, the height that corresponds to the base is perpendicular line segement from the opposite vertex to the base.

Three Dimensional Figures: Right Circular Cylinder

Axis is perpendicular to its bases. The height of a right circular cylinder is the perpendicular distance between the two bases. Because the axis of a right circular cylinder is perpendicular to both bases, the height of a right circular cylinder is equal to the length of the axis. Volume: V=Pie(r)^2(h) Surface Area: Sum of the areas of the two bases and the area of its lateral surface or A=2(Pie[r]^2)+2Pie(r)(h)

Ratio

Can be shown as x:x or x/x Equal Ration = Divide (need less) or multiple(if need more) each number in the ratio by the same number ex: if you have ratio 3:6 divide by three get 1:2 a *proportion* is an equation relating two ratios ex: 9/12=3/4 To solve proportion you *cross multiply* then *solve for x* ex: Find number x so that the ratio of x to 49 is the same as the ratio 3 to 21. x/49=3/21= 21x=(3)(49) x=(3)(49)/21 = 7

Linear Equations in Two Variables

Can be written ax+by=c When in this form easier to use the x|y graph by making x=0 and plugging it in and the y=0 and plugging it in BECAUSE you need to find the x & y-intercepts to graph this.

Convert Mixed Numbers to Fraction

Convert whole # to a fraction w/same denominator as fraction & then just add the two fractions ex: 4 3/8 = 4/1=4(8)/1(8)=32/8 + 3/8= 35/8

The Areas of The Special Quadrilaterals

For all parallelograms including rectangles & squares the Area (A) is given by the formula A=bh where b =length of a based h=is the length of the corresponding height

The Area of a Trapezoid

Formula: A=1/2(b1+b2)(h) where b1 & b2 are lengths of bases h= corresponding height

Percent Increase

Found by *dividing the amount of increase* (subtract the two numbers) *by the base* (initial number b4 change/starting number) ex: Quantity increases from 600 to 750 & initial amount is 600 so increase is 750-600=150 THEN 750-600/600 = 150/600 = 25/100 = 25%

Circles

Given point O in a plane & a positive number R the set of points in the plane that are a distance of R units from O is called a *circle*. The point O is called the *center* The distance R is called the *radius* The *diameter* of a circle is twice the radius. Two circles with equal radii are called *congruent circles* Any line segment joining two points on the circle is called a *chord* Radius - any line segment joining a point on the circle and the center of the circle Diameter - a chord that passed through the center of the circle. *Circumference* - The distance around a circle, which the like the perimeter of a polygon. The ratio of the circumference (C) to the diameter (D) is : Pie OR c/d=Pie Pie= 3.14 or 22/7 Circumference related to the radius: *C=2Pie(r)* *Angles Measure 360*

Three Dimensional Figures: Rectangle Solid or Rectangular Prism

Has 6 rectangular surfaces called *faces* (3D). Adjacent faces are perpendicular to each other. Each line segment that is the intersection of two faces is called an edge & each point at which the edges intersect is called a vertex. 12 edges and 8 vertices The dimensions of a rectangular solid are length (l) the width (w) and height (h) A rectangular solid with six squares faces is called a cube in which case l=w=h The *volume* (v) of a rectangular solid is the product os its three dimensions or *V=lwh* The *surface area* of a rectangular solid is the sum of the areas of the six faces or *A=2(lw+lh+wh)*

Adding & Subtracting Fractions

If *same denominator* just add/subtract If diff denominator must 1st find common denominator (common multiple of the 2 denominator) & convert so have same denominator. Then add or subtract top & keep the denominator. ex: 1/3-2/5 (if low numbers can just multiple 3x5=15) so get a common denominator of 15. 1(5) & 3(5) = 5/15 THEN -2(3)/5(3)=-6/15 so 5/15 + -6/15=-1/15

Compound Interest Cont.

If amount P is invested at an annual interest rate of r percent *compounded n times a year* then value V of the investment at end of T years is given by the formula

Reducing Fractions

If numerator & denominator have common factor the fraction can be reduced.

Applications Cont: Simple Interest

Interest earned on an investment during a specific time period can be computed as *simple interest* or *compound interest* Simple - based only on initial deposit, amount where interest is computed called *principal* for the entire time period. If amount P is invested at a simple annual interest rate of r percent, then value V of the investment at the end of T years is given by formula

The Pythagorean Theorem

In a *right triangle* the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. This can be used to find the length of one side of a right triangle if the lengths of the other two sides are known. Can also be used to find the ratios of the lengths of the sides of two special right triangles 1.) Isosceles Right Triangle (i.e. if you get x to x to square root of 2 this is the same as ratio 1 to 1 to the square root of 2). 2.)30-60-90 degree Right Triangle - which is half of an equilateral triangle (side opposite 30 degree angle is x, side opposite 60 degrees is x square root 3 and opposite 90 degree is 2x *Remember the opposite of squaring something (2^2) is the square root & vice versa*

Three Dimensional Figures

Include rectangle solids, cubes, cylinders, spheres , pyramids & cones.

Part 4 *Data Analysis* Distribution of a Variable/Distribution of Data

Indicates how frequently different categorical or numerical data values are observed in the data

Geometry: Polygons Convex Polygon

Is a closed figure formed by 3 or more line segments all of which are in the same plane The line segments are called *sides* of the polygon. Each side is joined to two other sides at its endpoints & endpoints are called *vertices* Polygon means convex polygon which means the measure of each interior angle is less than 180 degrees *The sum of the measures of the interior angles of an n-sided polygon is (n-2)(180 degrees)* ex: Hexagon has 6 sides so (6-2)(180) = 720

Constant

Is a number that has no variable attached and is ^ 0 unless otherwise noted

Degree

Is the exponent 4x^6 4 is the coefficient x is the variable 6 is the degree

Variables

Letter that represents a quantity whose value is unknown If written without an exponent has a degree of 1 x is really x^1

Tangent

Line that lies in the same plane as the circle & intersects the circle at exactly one point called the *point of tangency*. If a line is tangent to a circle, then a radius drawn to the point of tangency is perpendicular to the tangent line.

Geometry: Perimeter of Polygon Area of Polygon

Perimeter - sum of the lengths of its sides Area - the area of the region enclosed by the polygon

Mean of Frequency Distribution

Mean = Sum of each frequency * midpoints/sum of all frequencies 1st.) Find midpoint of each group of frequencies by adding the two numbers & dividing by 2 (can also just find first midpoint the subtract the first number in the next midpoint by the first number in the last midpoint and take that number and add it on to the first midpoint = midpoint for the second group - continue that process). 2nd.) Multiply each midpoint by its corresponding frequency & add up all those answers 3rd.) Add up all the frequencies and divide the sum of all the frequencies *midpoints by the total number of frequencies

Negative Exponents

Negative Exponents: Negative # raised to even power= always positive Negative # raised to odd power = always negative a^-1= 1/a a^-2= 1/a^2 a^-3= 1/a^3 ETC.

Square Root Four Rules

Non-negative number that is squared by itself is that number ex: Square root of 16 is 4 because 4^2=16 (can also be -4 because [-4]^2=16 All positive numbers have two square roots; one positive & one negative Square root of Zero is ZERO Square Root is the opposite of Squaring something & vice versa Rule 1: (√a)^2=a Rule 2: √a^2=a Rule 3: √a√b=√ab Rule 4: √a/√b= √a/b

Composite Number

Not a prime number

Part 3: *Geometry* Line Segment Midpoint

Part of the line that contains the two points & all points between them - the two points are called *endpoints* Line segments have equal lengths that are called *congruent line segments* *Midpoint* - point that divides a line segment into two congruent line segements

Three Propositions to Determine if Two Triangles are Congruent

Proposition 1: *Side-Side-Side* (SSS) If three sides of one triangle are congruent to the three sides of another triangle. Proposition 2: *Side-Angle-Side* (SAS) If two sides & the included angle of another triangle are congruent to two sides & the included angle of another triangle. Proposition 3: *Angle-Side-Angle* (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. Also if *two angles of one triangle are congruent to two angles* of another triangle then the remaining angles are also congruent sine it has to add up to 180 degrees

Median of Frequency Distribution

Put in a graph with a group, Frequency, & Cumulative Frequency label. The first frequency is just that number. THEN add the first frequency to the next frequency to get the second frequency. Take that cumulative frequency and add it to the next frequency to get the third cumulative frequency (continue this). Take the last cumulative frequency (sum of the frequencies) & divide by 2. That answer is the score where the median is so match that score up with where it would be in the groups in the cumulative frequency column

Solving Quadratic Equations: 1.) Quadratic Formula

QE in the variable x is an equation that can be written in the form *ax^2+bx+c=0* where a is not equal to 0 Have no, one or two solutions One way to solve these is to use the *Quadratic Formula* Once you get the equation simplified you have to add & subtract the remaining numbers so you get two answers. You're solution *CAN'T BE ZERO* if it is then that is not a solution. Also you can't square root negative numbers so if you get that there is *NO SOLUTION*

Sector of a Circle Area of a sector

Region bounded by an arc of the circle & two radii Area= The ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degree measure of its arc to 360 degrees *s/area of the circle=degree of central angle/360

Coordinate Geometry: Graphing *Two-Variable* Linear Equations

Solve for y to get slope intercept form - then can plot y-intercept or make X/Y chart & fill in # for x to find y values (make up starting with 0 etc.)

Frequency Distribution

Table or graph that presents the categories/numerical values along with their corresponding frequencies.

Relative Frequency Distribution

Table/graph that presents the relative frequencies of the categories or numerical values.

Quotient

The answer (= _ ) The first # when 2 numbers can't be evenly divided into each other

Relative Frequency

The corresponding frequency divided by the total number of data. Relative frequencies may be expressed in terms of percents, fractions, or decimals.

Coordinate Geometry: Graphing Quadratic Equations; Parabola

The graph of a quadratic equation of the form y=ax^2+bx+c where a, b & c are constants and a is NOT 0 is called a *parabola* X-intercepts of the parabola are the soltions of the equation If a is *positive* the parabola opens *upward* and the vertex is its *lowest* point If a is *negative* the parabola opens *downward* and the vertex is its *highest* point *y-intercepts* is the y-coordinate where x=0 and you plug in zeros for x. The x & y's will be each others opposites on the graph & the symmetry line is the vertex (x-coordinate)

Coordinate Geometry: Graphing Circles

The graph of an equation of the form = *(x-a)^2 + (y-b)^2=r^2* is a *circle* with it's center at the point (a,b) and with radios r>0 Area for Circles = Pie(r)^2

Coordinate Geometry: Slope

The slope of a line passes through two points (x1, y1) and (x2, y2) where x1 is not equal to x2 and is defined = (called rise over run) Two lines are *parallel* if their slopes are equal Two lines are *perpendicular* if slopes are negative reciprocals of each other

Multiplying & Dividing Fractions

To *Multiply* the two numerators & then multiply the two denominators To *Divide* flip the second fraction (reciprocal) then multiply the two fractions.

Percents

To find *whole & part* divide *part/whole x 100* ex: What % of 150 is 12.9 12.9/150=0.086 > 8.6% To find *part that is certain percent of a whole* you either *1.) Multiply the whole by decimal of percent* (i.e. 30% of 350= 350 x 0.3=105) OR *2.) Find parts of 350 that yield same ratio as 30 parts out of 100* (i.e. x/350=30/100 . . x=30 x350/100=105) To find *whole* when given *percent & part* you either *1.) Use decimal equivalent* (i.e. 15 is 60% of what # . . 0.6z=15 because 60% of some # is 15, then divide both sides by 0.6 so z=15/0.6=25) *2.)Use proportion* (i.e. 60/100 so 15/x=60/100 . . 60z=15 x 100 . . z=15 x 100/60=1,500/60=25

Coordinate Geometry: Graphing Functions

To graph a function in the xy-plane you represent each input x and its corresponding output f(x) as a point (x,y), where y=f(x). You use x-axis for the input & the y-axis for the output. The *linear function* defined f(x)=-1/2x+1 graphed in the xy-plane is the line with the *linear equation* y=-1/2x+1 Contents of square root are always quadratic so square equation on both sides

Applications

Translating verbal descriptions into algebraic expressions. Include averages, mixture, & work problems. distance=rate time=hours distance/output=(r)(t) (r=rate or speed) rate=distance or output/time time= output or distance/rate

Three Dimensional Figures: Circular Cylinder

Two bases that are congruent circles lying in parallel planes and a lateral surface made of all line segments joining the centers of the two circles. The latter line segment is called the axis of the cylinder.

Geometry: Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines

Geometry: Perpendicular Lines Right Angle

Two lines that intersect to form four congruent angles are perpendicular lines Each of the four angles has a measure of 90 degrees & an angle with a 90 degree measure is called a *right angle*

*Coordinate Geometry* (still apart of Algebra section): Rectangular Coordinate System

Two real # lines that are perpendicular to each other & that intersect at their respective zero points = also called *xy-coordinate system* or *xy-plane* x-axis=horizontal line y-axis=vertical line

Congruent Triangles

Two triangles that have the same shape & size are called *congruent triangles*. They are congruent if their vertices can be matched up so that the corresponding angles & the corresponding sides are congruent. If you are given PQR for one triangle & STU for another it tells you angle P=S, Q=T & R=U

Similar Triangles

Two triangles that have the same shape but not necessarily the same size. Two triangles are similar if their vertices can be matched up so that the corresponding angles are congruent, or the lengths of the corresponding sides have the same ratio which is called *the scale factor of similarity*. *This is all 30-60-90 degree right triangles

Special Triangles

Type 1 = Triangle with three congruent sides is called an *equilateral triangle*. The measures of the three interior angles are equal & each measure if *60 degrees* Type 2 = Triangle with at least two congruent sides is called an *isosceles triangle*. If triangle has 2 congruent sides then the angles opposite the two congruent sides are congruent. Type 3= Triangle with an interior right angle is called a *right triangle*. The side opposite to the right angle is called the *hypotenuse*, the other two sides are called legs.

Four types of Special Quadrilaterals

Type 1: Quadrilateral with four right angles is called a *rectangle*. Opposite sides of a rectangle are parallel & congruent and the two diagonals are also congruent. Type 2: Rectangle with four congruent sides is called a *square*. Type 3: Quadrilateral in which both pairs of opposite sides are parallel is called a *parallelogram* Here opposite sides are congruent & opposite angles are congruent. *Note all rectangles are parallelograms. Type 4: Quadrilateral in which at least one pair of opposite sides is parallel is called a *trapezoid*. Two opposite, parrallel sides of the trapezoid are called *bases* of the trapezoid.

Coordinate Geometry: Calculating the Distance between two Points

Use *Pythagorean Theorem* To find length construct right triangle w/hypotenuse (long slanted side) by drawing vertical line down from R & horizontal line to right until they intersect. Then count how long the two sides are and use a^2+b^2=c^2 when you get a^2 +b^2 you square/add them together and then you will have to find the square root of that number (since the opposite of squaring something i the square root).

Two Methods of Solving Linear Equations: 2.) Elimination

Want to make coefficients of one variable the same in both equations so one variable can be *eliminated* by either adding/subtracting one from another ex: 4x+3y=13 x+2y=2 4(x+2y)=4(2) - want to multiply both sides of second equation by 4 so you have two equations with same coefficient of x 4x+3y=13 & 4x+8y=8 *Subtract* 4x+3y=13 - 4x+8y=8 -5y= 5 /-5 /-5 y=-1 *substitute* -1 for y in either equation 4x+3(-1)=13 4x-3=13 +3 +3 4x=16 /4 /4 x=4

Factorization

When integers are multiplied each multiplied number is a *factor* OR *divisor* of the resulting product Have to include neg & pos in factors of a number example: 4 has six factors = -4, -2, -1, 1, 2, 4

Geometry: Angles Opposite Angles

When two lines intersect at a point they form *angles* The *sum* of the measures of *four angles* is 360 degrees. Opposite angles have equal measure & angles that have equal measures are called *congruent angles*

Coefficient

number that is multiplied by variables ex: 2x 2 is the coefficient x is the variable

Ordered Pair

numbers (x, y) that make equation true when values of x & y are plugged into the equation usually only one ordered pair that satisfy both equations but possible for there to be no solutions or infinite solutions

Higher Order Roots

numbers like 3 that are written differently i.e. 3 (cube root) is written as 3√n = what # multiplied 3 times is = n

Prime Divisors

are called prime factorization - where you do the factor tree until you have all prime numbers: 12 / \ 2 6 / \ 2 3 SO = (2^2) (3)

Absolute Value

distance between a number (x) and zero on the number line Looks like

Decimals

each place value is a power of 10

Greatest Common Factor

greatest positive integer that is a divisor of both numbers. ex: GMC of 30 & 75 is 15 because. . . divisors of 30 are 30/1=30, 30/2=15, 30/3=10, 30/4=7.5 (four not a divisor since not even), 30/5=6, 30/6=5, 30/10=3, 30/15=2 75 are: 1, 3, 5, 15

Applications Cont: Compound Interest

interest is added to the principal at regular intervals (annually[yearly]/quarterly[3 months]/monthly). Each time interest is added to the principal, the interest is compounded. After each compounding, interest is earned on the new principal, which is sum of the preceding principal & the interest just added. If amount P is invest at an annual interest rate of R percent, *compounded annually* then value of V of the investment at the end of T years is given by the formula:

Least Common Multiple

least positive integer that is a multiple of both numbers Ex: LCM of 30 & 75 is 150 because multiples of 30 are: 30x1=30, 30x2=60, 90, 120, 150 75 are: 75x1=75, 75x2=150, 225 150 is the least multiple there

Remainder

number left over from the same 2 numbers that can't be evenly divided (quotient). Find this by finding the greatest multiple of the denominator that is less than or equal to the numerator. Then subtract that multiple from the numerator. Ex: 100/45=2.222 so quotient is 2 and the greatest multiple of 45 that is less than/equal to 100 is 45x2=90 SO 100-90=10 The answer is *2 remainder 10* *also be careful with negatives (15 is more than 13 BUT -15 is less than -13).

Vertex Formula

x=h=-b/2(a) THEN to find y value just plug above into formula for the x's

Point Slope Form

y-b=m(x-a) Can be used to be converted into slope intercept form

Coordinate Geometry: Graphing Linear Equations Steps

y=mx+b which is a straight line in the xy-plane where m=slope of line b=y-intercept *x-intercepts* of a graph are the x-coordinates of the points where graph intersects *x-axis* & *y-intercepts are the y-coordinates where the graph intersects *y-axis* Sometimes x-intercept/y-intercept refer to actual intersection points *Step 1* = Put equation in Slope Intercept form (y=mx+b) - may have to simplify expression to get into slope intercept form. If slope is positive # then slope of line will be positive slope (going up from left to right) The # in the b position is the *y-intercept* which is where the graphed line will cross the y-axis *Step 2* = Graph the y-intercept point In equations without a # in b or no b you just plot first point at (0,0). *Step 3* = From point plotted at y-axis you use slope to find second point (a variable x counts as one so if you have y=x-3 the slope would be 1). If in fraction form like y=2/3x + 5 you would plot (0,5) on y-axis and then move up 2 & over three for slope x-intercepts to find set y=0 & solve for x y-intercepts can set x variables to 0