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Period

In a large number, periods are groups of 3 digits separated by commas or by spaces.

Amplitude

how tall it is

Domain

The ______________________ is all the possible INPUT, X, values.

Range

The difference between the greatest number and the least number in a set of data.

a

amplitude

vertical stretch

the parabola appears narrower when y= ax^2 and the absolute value of a is greater than 1

vertical shrink

the parabola appears wider when y= ax^2 and the absolute value of a is less than 1

a is negative

horizontal flip on x-axis

b

period

more cycles

b > 1

less cycles

b < 1

b is negative

vertical flip across y-axis (in cosine, no visual change)

increments

Evenly spaced units; the axis of a graph must use proper _______________

Vertical shift

# is outside ( ) that is added or subtracted

shift up

k is positve

shift down

k is negative

Horizontal shift

# is inside ( ) that is being added or subtracted.

shift left

F(x+4) does what?

shift right

F(x-5) does what?

Asymptote

An imaginary line on a graph that acts as a boundary line.

Crest

Highest point of a wave. (maximum y-value)

Intercepts

The point(s) at which the graph intersects or touches the x-axis or y-axis.

Periodic Functions

Functions that repeat their values over a regular interval.

Period or Wavelength

The distance required for a function to complete one full cycle. The Greek letter λ (lambda) is used to represent wavelength. (Period = 2π/b)

Transformation

a movement on the cartesian plane

Trough

Lowest point of a wave. (minimum y-value)

Undefined

when dividing a nonzero number by zero,

La Buena Onda

The graphs formed by sine and cosine functions...more commonly referred to as waves.

Cosine

The ratio of the adjacent side to the hypotenuse of a right triangle which means adjacent over hypotenuse,

sine

The ratio of the opposite side to the hypotenuse of a right triangle which means opposite over hypotenuse,

Tangent

The ratio of the opposite side to the adjacent side of a right triangle which means opposite over adjacent.

reciprocal

One of two numbers whose product is 1; it is also called multiplicative inverse.

horizontal stretch

All y-coordinates are multiplied by a number less than 1. The points get closer to the x-axis (graph gets fatter).

absolute value

The distance of a number from zero on a number line; shown by | |

Parent (mother) Function

The original or most basic form of a function.

slope

(y₂-y₁)/(x₂-x₁)

Trigonometric Ratios

The comparison of two sides of a right triangle.

Domain of sin(x)

All real numbers

Domain of cos(x)

All real numbers

Domain of tan(x)

D = {x: x = R, except π/2 + nπ where n is an odd integer}

Domain of csc(x)

D = {x: x = R, except nπ when n is an integer}

Domain of sec(x)

D = {x: x = R, except π/2 + nπ when n is an odd integer}

Domain of cot(x)

D = {x: x = R, except nπ when n is an integer}

Range of sin(x)

R = {y: -1 ≤ y ≤ 1}

Range of cos(x)

R = {y: -1 ≤ y ≤ 1}

Range of tan(x)

All real numbers

Range of csc(x)

R = {y: y ≤ -1 or y ≥ 1}

Range of sec(x)

R = {y: y ≤ -1 or y ≥ 1}

Range of cot(x)

All real numbers

sin, cos, csc, and sec

Which trigonometric values is 2π a period for?

tan, cot

Which trigonometric values is π a period for?

Quadrant I

positive, positive, positive

quadrant II

negative, positive

Quadrant III

negative, negative, negative

Quadrant IV

positive, negative

Pythagorean identities

sin²θ + cos²θ = 1

sin(-θ)

-sinθ

cos(-θ)

cosθ

tan(-θ)

-tanθ

cot(-θ)

-cotθ

sec(-θ)

secθ

csc(-θ)

-cscθ

sin(x) = cos(x - π/2)

Relationship of the graph of sin and the graph of cos

2π/b

Find period for y = a*sin(bx + c) + d and y = a*cos(bx + c) + d

c/b

Find phase shift for y = a*sin(bx + c) + d and y = a*cos(bx + c) + d

Find the start and end points of a cycle of sin/cos

Set bx + c = 0 and bx + c = 2π

y = cos⁻¹(x)

x = cos(y)

sin⁻¹(sin x) = x

where -π/2 ≤ x ≤ π/2

sin(sin⁻¹ x) = x

where -1 ≤ x ≤ 1