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Level 441

Graphing Trigonometric Functions


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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