Math 1314 Algebra Sec. 1.1-1.6

ordered pair

(x,y) gives the coordinates of the location of a point

factoring trinomial (a≠1)

*SIMPLIFY FIRST* 1. Make 2 sets as in (a=1), but will use trial and error with multiplication table.

factoring trinomials (a=1)

*SIMPLIFY FIRST* 1. Make 2 sets of parentheses, put factors of x² in 1st position of each set. 2. 2nd positions are the factors of c that add to b.

factoring difference of squares

*SIMPLIFY FIRST* a²-b²=(a+b)(a-b) 1. Recognize perfect squares.

polynomial

*Terms are added or subtracted and contain powers of (x).

slope

*represents the distance up and down you must move to get to another point ÷ distance left or right that you must move to get to another point.

radical

1. when simplifying (for example √27x³), you would take out your perfect square and along with it, the any even number of exponents you can take out (√9x² √3x

function

A correspondence between two sets, called domain and range, in which each member of the domain is assigned to exactly one element of the range. Domain: x-values Range: y-values

linear function

A function is called linear if for each ordered pair contained in the function, the rate of change is constant from pair to pair. The graph of a linear function is a straight line.

slope-intercept form

A linear function can be written in slope-intercept form, f(x) = mx + b, where m and b are constants. Specifically, m = slope (rate of change) of the line and (0, b) is the y-intercept of the line.

expression

A mathematical statement with variables and constants but which does not contain any relation symbol (<,>,=,...) Ex: 2y + 7 We simplify expressions.

linear regression

A method of "curve fitting" that can be done on a graphing calculator to approximate a linear equation of dat that is not necessarily completely linear, but very close to linear.

exponent

A number that tells how many times the base should be multiplied by itself

distributive properties

A property of real numbers that states that the product of a number and the sum or difference of two numbers is the same as the sum or difference of their products. Example: 2( 3 + x) = 2(3) + 2(x)

commutative property

A property of real numbers that states that the sum or product of two terms is unaffected by the order in which the terms are added or multiplied; i.e., the sum or product remains the same. . Ex: 2x + y = y + 2x

correlation coefficient

A value of "r" on a calculator. Tells how accurate the model is. This value needs to be "close" to either 1 or -1 for the linear approximation to be accurate. If r=1, the data is exactly linear with a positive trend (slope is positive). If r=-1, the data is exactly linear with a negative trend (slope is negative).

integer

All whole numbers (both positive and negative) and zero.

quadratic equation

An equation that can be written in the standard form ax²+bx+c=0

inverse operation

An operation that undoes another operation. For example, multiplication is the inverse operation for division, addition is the inverse operation for subtraction.

slope of horizontal line

Answer: 0

slope of vertical line

Answer: undefined

quadratic equation

Equation where the highest power of x is 2 EX: x²+7x+10

vertical-line test

For a graph to be classified as a function, any vertical line must cross the graph at most one time. If some vertical line crosses the graph more than once, then the graph does not represent a function.

midpoint formula (M.P.)

If the endpoints of a segment are (𝑥₁, 𝑦₁) and (𝑥₂, 𝑦₂), then the coordinates of the midpoint of the segment are given by M.P.=(x₂+x₁)/2) , ((y₂+y₁)/2)

|

In set-builder notation means "such that."

distance formula

The distance, d, between any two points (𝑥₁, 𝑦₁) and (𝑥₂, 𝑦₂) is given by d=√(x₂-x₁)²+(y₂-y₁)²

degree of a term

The exponents on the variables added together EX: (-2x²y³= degree of 5)

coefficient

The numerical factor of a term in a polynomial. EX: 14 is the coefficient in the term 14x

integer

The set of positive whole numbers and their opposites (negative numbers) and 0. Numbers that are NOT fractions or decimals.

slope

The slope represents the "rate of change" or "steepness" of a linear function.

slope of parallel lines

The slopes are equal EX: y = 2x + 5 and y = 2x - 7

quadratic

The solution of a quadratic equation in the form ax²+bx+c=0, where a≠0, are given by the formula.

equation of a circle

The standard form of the equation of a circle with center (h, k) and radius r is given by (x-h)² + (y-k)² = r²

order of operations

The steps used to evaluate a mathematical expression. First, evaluate expressions inside grouping symbols (parentheses & brackets). Second, evaluate powers (exponents). Third, multiply and divide left to right. Last, add and subtract left to right.

no solution problems

This happens when you solve and yield an equation that is always false. EX: 3=7

all real numbers are a solution

This happens when you solve and yield an equation that is always true. EX: 130=130

∪

Union (all together) ("OR") (joining) EX: (-∞,-2] ∪ (5,∞)

plug and chug formula to find slope

Use if you are asked to find the slope given two points: m = y₂ - y₁ ÷ x₂ - x₁

practical formula to find slope

Use if you are finding slope from a graph or wishing to graph by using slope: m = rise ÷ run

FOIL

When multiplying binomials: First, Outer, Inner, Last

Standard Window

[-10, 10, -10, 10] Xscl = 1, Yscl = 1.

polynomials

a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power arranged in the degree of term EX: (3x³ + y² - x + 4) = 6th degree polynomial

parabola

a u-shaped graph of a quadratic function

domain

all x values that correspond to points on a graph

range

all y values that correspond to points on a graph

dividing inequalities

always change direction of the symbol (≥,≤,>,<)

binomials

an expression consisting of two terms connected by a plus sign or minus sign; polynomial with two terms.

interval notation

example: [-4, ∞)

set notation

example: {x|x≥-4}

term

mathematical expressions made of a coefficient (EX: 2, 99, 1.5) and a variable (EX: x, ab², x³y²) EX: 2x², 3y

x-intercept

point(s) at which a graph crosses the x-axis

y-intercept

point(s) at which a graph crosses the y-axis

trinomials

polynomials with three terms EX: (a + bx - cx²)

slope of perpendicular lines

the slopes are opposite in sign and reciprocals of each other EX: y = 3x + 2 and y = -1/3x -4

zeros of a function

the x-values for which f(x)=0; the x-values where the function crosses the x-axis.

constant term

value of a term that always stays the same EX: 3x² - 5, 5 is the constant

vertical line equation

x = # (constant)

expanded form

x²-2x-24 not (x+4)(x-6)

horizontal line equation

y - # (constant)

Factored Form

y = (x - 2)(x +3); shows the roots of a parabola

slope-intercept equation

y = mx + b, where m=slope and b=y-intercept