Pre-Cal CH 1.1 and 1.2

difference of functions is given by

(f - g)(x) = f(x) - g(x). CHANGE SIGNS (+),(-) IN g(x)!!!!!

composite function, f ◦ g, is defined as

(f ◦ g) = f(g(x))

circle function

(x-h)² + (y-h)² = r² equations are in the form x² + y² = r²

circle function

(x-h)² + (y-k)² = r² where (h,k)=center - equations are in the form x² + y² = r²

circle equation

(x−h)²+(y−k)²=r² where (h,k)=center

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.

fraction rules (1.2,15)

1. Add and Subtract: Find least common denominator, add or subtract nominators and denominator stays same. EX: 3/4 + 2/3 = 9/12 + 6/12 = 15/12 = 5/4. 2. Multiply: Multiply numerators and multiply denominators. EX: 3/4 ×3/4 = 9/16. 3. Divide: Keep top fraction, chg ÷ to ×, flip bottom fraction. EX: 3/4 ÷ 2/3 = 3/4 × 3/2

equation of a circle: how to find (1.1,8)

1. Find the center (use the midpoint formula if given the endpoints of the diameter). 2. Find the radius (use the distance formula - if finding diameter first, be sure to divide by two to determine the radius). 3. Plug the center, (h, k), and the radius, r, into the equation (𝑥𝑥 − ℎ)2 + (𝑦𝑦 − 𝑘𝑘)2 = 𝑟𝑟2 and simplify. Leave your answer in standard form.

calculator: graphing equations (1.1,5)

1. Solve the given equation for y (if not already solved for y). 2. Type this equation into the "y=" menu. 3. Set an appropriate window so that x and y intercepts are clearly visible. Hint: The constant value (value without a variable) of an equation solved for y, is the value of the y-intercept. This value will be a hint to how to set your window. 4. Press the Graph key.

evaluating functions (1.2,14)

1. Substitute the given value in for every x in the expression. 2. Simplify completely.

distance formula

= √(x₂-x₁)² + (y₂-y₁)²

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. Two domains may by assigned to one range, but one domain may not be assigned to two range, and not all ranges need to be used. Domain: x-values Range: y-values

absolute value function

A function containing the |x|. It is shaped like the letter v

linear function

A function with a constant rate of change; often in the form y = mx + b or y = Ax + By = C

expression

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

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).

Horizontal Shifts

A value that is added or subtracted "inside" the function. NOT consistent with the sign (+ means go LEFT, - means go RIGHT).

range

All possible output values (y values)

input

An element of the domain of a function.

output

An element of the range of a function

Difference of Squares

Both term in front and back are square terms and sign must be (-). EX: x2 - 9 = (x + 3)(x - 3)

graphing functions by plotting point (creating t-chart)

EX: Graph f(x) = -x²

Fraction Exponent Rule

EX: X^a/b = a√x^b

Complex Conjugate

EX: a + bi and a - bi or (4 + 8i)(4 - 8i)

Finding LCD Help

Factor the denominators AND multiply the denominators together to find LCD.

verticle line test

If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function.

|

In set-builder notation means "such that."

Absolute Value

Means how many units from 0 on a number line. By definition must equal a positive number.

axes

Perpendicular number lines that intersect at (0,0)

distance formula (1.1,6)

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)

abscissa

The first number of an ordered pair. Example: The 2 in (2,8).

first coordinate

The first number of an ordered pair. Example: The 2 in (2,8).

x-coordinate

The first number of an ordered pair. Example: The 2 in (2,8).

ordered pair

The location of a single point on a rectangular coordinate system, where the digits represent the position relative to the x-axis and y-axis [e.g., (x, y) or (1, -2)].

coefficient

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

ordinate

The second number of an ordered pair. Example: The 8 in (2,8).

second coordinate

The second number of an ordered pair. Example: The 8 in (2,8).

y-coordinate

The second number of an ordered pair. Example: The 8 in (2,8).

equation of a circle formula (1.1,7)

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

x-intercept

The value of x at the point where a line or graph intersects the x-axis. The value of y is zero (0) at this point.

y-intercept

The value of y at the point where a line or graph intersects the y-axis. The value of x is zero (0) at this point.

all real numbers are a solution

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

Dividing Complex Numbers

To DIVIDE complex numbers, always multiply numerator and denominator by the CONJUGATE of the DENOMINATOR. The goal is to eliminate i from the denominator.

∪

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

Imaginary Number

We define √−1 = i. We can also say that i = −1.

FOIL

When multiplying binomials: First, Outer, Inner, Last

midpoint formula

[(x₁ + x₂)/2, (y₁ + y₂)/2]

Cartesian Coordinate System

a coordinate system is which the coordinates of a point and its distances from a set of perpendicular lines that intersect at an origin

relation

a correspondence between a first set (domain) and a second set (range), such that each member of the domain corresponds to at least one member of the range

function

a correspondence between a first set (domain) and a second set (range), such that each member of the domain corresponds to exactly one member of the range.

Complex Number

a number in the form a + bi, where a and b are real numbers. The number a is called the real part and the number b is called the imaginary part.

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.

reciprocal function

f(x) = 1/x where x ≠ 0, the x- and y-axis are asymptotes - domain and range are set of all nonzero real numbers - no intercepts - decrease on (-∞,0) and (0,∞) - odd function

cubic function

f(x) = x³ equations are in the form y=x³

cubic function

f(x) = x³ equations are in the form y=x³

finding function values given a graph (1.2,17)

f(x)=y, If given x, find where x intersects with y. (pic)

graph is of a hyperbole

horseshoe shape

A relation is said to be "symmetric about the origin"

if for every point (x, y) on the graph, there also exists a point (-x,-y ). Also called ODD

A relation is said to be "symmetric about the y-axis"

if for every point (x, y) on the graph, there also exists a point (-x,y ). Also called EVEN.

A relation is said to be "symmetric about the x-axis"

if for every point (x, y) on the graph, there also exists a point (x,-y ).

y-axis

indicates the point's vertical location with respect to the x-axis

x-axis

indicates the point's vertical location with respect to the y-axis

decreasing function

the function drops from left to right (slopes "downhill").

increasing function

the function rises from left to right (slopes "uphill").

constant function

the function values remain the same from left to right (slope is fixed).

origin

the point where the x-axis and y- axis number lines intersect (0,0)

quadrants

the two axes divide the plane into four sections labeled with Roman numerals (I, II, III, IV) starting at the positive x-axis and going around counter-clockwise

plot

to graph an ordered pair or equation

to graph an equation

to make a drawing that represents the solutions of that equation

constant term

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

domain

x-coordinates or values in a data set

independent variable

x-coordinates or values in a data set The information is independent and does not rely on another variable to work Ex: time

independent variable

x-coordinates or values in a data set The information is independent and does not rely on another variable to work.

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

Basic Reciprocal Shape

y = 1/x

quadratic function

y = ax²+bx+c equations are in the form y=x² graph is of a hyperbole

Basic Linear Shape

y = x

Basic Quadratic Shape

y = x2

Basic Cubic Shape

y = x3

Basic Absolute Value Shape

y = |x|

Basic Square Root Shape

y = √x

dependent variable

y-coordinates or values in a data set The information is dependent on another variable Ex: distance (relies on time)

dependent variable

y-coordinates or values in a data set The information is dependent on another variable

square root function

y=a(√x-h)+k equations are in the form y=√x

logarithmic function

y=logbx equations are in the form y=bⁿ The y-axis is an asymptote. Inverse of an exponential function.