Math 4-8 115

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Reflection over x-axis

(-x,-y)

Associative Property of Addition

(a + b) + c = a + (b + c)

Associative Property of Multiplication

(a x b) x c = a(b x c)

Distributive Property of Multiplication over Addition

(a) x (b + c) = (a x b) + (a x c)

slope

(y₂- y₁) / (x₂- x₁) Slope = 0, horizontal line Slope is 1/0 = vertical line Slope is negative = line is downward Slope is positive = line is upward

function

1. A relationship from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. 2. The action or actions that an item is designed to perform.

Surface area of a sphere

4πr²

linear function

A function whose graph is a straight line. concepts include slope, domain, range and intercepts

vertex

A point where two or more straight lines meet.

Parallelogram

A quadrilateral with two pairs of parallel sides

Proportion

A relationship between two quantities that dictates how one changes when the other changes

horizontal shift

A rigid transformation of a graph in a horizontal direction, either left or right. f (x-b)

recursive formula

A rule for a sequence in which one or more previous terms are used to generate the next term

Prism

A solid figure that has two congruent, parallel polygons as its bases. Its sides are parallelograms

rectangular prism

A solid shape that has: 6 faces (4 rectangles & 2 squares), 8 vertices (corners), and 12 equal edges.

sphere

A solid shape that is perfectly round like a ball. No faces, edges, or vertices.

pyramid

A solid shape with a polygon as a base and triangular faces that come to a point (vertex or apex)

vertical line test

A test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation

cube

A three-dimensional shape with six square faces, or sides, with three meeting at each vertex

isosceles triangle

A triangle that has 2 equal sides.

acute triangle

A triangle with 3 acute angles

Outlier

A value much greater or much less than the others in a data set

prime number

A whole number that has exactly two factors, 1 and itself.

Area of a triangle

A=1/2bh

Area of a rectangle

A=bh

Area of a circle

A=πr²

vertical translation

An up or down slide; the y-coordinate changes.

standard form of a linear equation

Ax + By = C, where A,B, and C are real numbers, and A and B are not both zero.

VLT

Example 1 - y = x2 a. Choose values of x: -2, -1, 0, 1, and 2b. Substitute these values into the equation, y = x2c. The corresponding y-values: 4, 1, 0, 1, 4d. Graph the points (-2, 4), (-1, 1) , (0, 0), (1, 1), and (2, 4)

Hypotenuse Leg

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Rational numbers

Includes all integers, decimals, and fractions. Any terminating or repeating decimal is a rational number.

irrational numbers

Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating.

Substitution method

Replacing one variable with an equivalent expression containing the other variable

Surface Area of a Cylinder

SA=2πr²+2πrh

Quadratic Function Equation

Solve for x: x^2 - 25 = 0 a. Isolate x, we have x^2 = 25 b. Take the square root of both sides, x = + or - 5 c. The quadratic function y = x^2 crosses the x-axis at +5 and -5 d. Therefore, +5 and -5 are the solutions

circle

The set of all points in a plane that are the same distance from a given point called the center

Domain

The set of the first numbers of the ordered pairs in a relation

least common multiple (LCM)

The smallest multiple (other than zero) that two or more numbers have in common.

complex conjugate

Two complex numbers of the form a + bi and a - bi

Multiplicative Inverse

Two numbers with a product of 1

Volume of a cube or Prism:

V= L x W x H or S^3

Volume of a sphere

V=4/3πr³

Volume of a cylinder/prism

V=Bh

Volume of a rectangle

V=lwh

Volume of a rectangular prism

V=lwh

Volume of a cylinder

V=πr²h

Constant by proportionality

When two quantities have a proportional relationship,

Commutative Property of Addition

a + b = b + a

inscribed circle

a circle in which each side of the polygon is tangent to the circle

Discriminant

a function of the coefficients of a polynomial equation whose value gives information about the roots of the polynomial.

exponential form

a function of the form in which the input variable x occurs as an exponent

nonlinear function

a function whose graph is not a line or part of a line

Population

a group of organisms of the same species populating a given area

Asymptote

a line such that the distance between the curve and the line approaches zero as they tend to infinity

z-score

a measure of how many standard deviations you are away from the norm (average or mean)

complex number

a number of the form a + bi where a and b are real numbers and i is the square root of -1

The decimal, or base 10, system

a number system that uses ten different digits

imaginary number

a number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = -1

Parabola

a plane curve formed by the intersection of a right circular cone and a plane parallel to an element of the curve

quadrilateral

a polygon with four edges and four vertices or corners

rectangle

a quadrilateral with four right angles

square

a quadrilateral with four right angles and four congruent sides

inverse proportion

a relationship in which an increase in one quantity is accompanied by a decrease in another quantity.

arithmetic series

a sequence of numbers such that the difference between the consecutive terms is constant

system of equations

a set of two or more equations containing the same variables

scalene triangle

a triangle with no congruent sides

scientific notation

a way of writing numbers using exponents

Commutative Property of Multiplication

a x b = b x a

Solve for x: 2x^2 + 5x - 3 = 0

a. Use the factoring method b. By expressing 2x^2 + 5x - 3 = 0 as a product of two binomials, we get (2x -1) (x + 3)2x -1 = 0 and x + 3 = 0 c. The two solutions are x = 1/2 and x = -3 d. the function y = 2x^2 + 5x - 3 crosses the x-axis at 1/2 and -3, which are two solution

Percent to Decimal

a. move the decimal point two places to the left b. remove the percent sign Ex: 325% = 3.25

decimal to percent

a. move the decimal point two places to the right b. add a percent sign Ex: 0.7 = 70%

Fraction to Percent

a. multiply by 100 b. add the percent sign Ex: 3/8 = (3/8 x 100%) = 37.5%

percent to fraction

a. remove the percent sign b. place number over 100 c. reduce answer to its simplest form Ex. 1.5% = 1.5/100 = 15/1000 = 3/200

vertical stretch/shrink

af(x)

Integer

any of the natural numbers (positive or negative) or zero

relation

any set of ordered pairs (x, y)

quadratic equation in standard form

ax^2+bx+c=0

Geometric Series Formula

a↓n = a↓1 • r^(n-1), r = common ratio a series with a constant ratio between successive terms

Congruent Triangles/Polygons

corresponding parts of congruent triangles (polygons) are congruent

Another form of a quadratic function

f(x) = a(x - h)^2 + k where a, h and k are constants The vertex of the parabola is given by the point (h, k)

graph of a quadratic function

f(x) = ax^2 + bx + c creates a curve called a parabola

Graphing method

graph the equation and the solution is the x-intercept, where the parabola meets or crosses the x-axis

triangle

is a polygon with three edges and three vertices

Elimination method

solving systems by adding or subtracting equations to eliminate a variable

mean

the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores

greatest comon factor

the largest number that is a factor of two or more numbers

Median

the middle score in a distribution; half the scores are above it and half are below it

Real Numbers

the set of all rational and irrational numbers

range

the set of second numbers of the ordered pairs in a relation

vertical line

undefined slope

Binary, or base 2

uses only the numbers 0 and 1

slope-intercept form

y = mx + b

horizontal line

y=b where b is the y-intercept of the line slope = 0

horizontal stretch/shrink

y=f(ax)


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Chapter 7: Founding a Nation, 1783—1791

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