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Questions and Answers
What distinguishes a periodic function from other types of functions?
What distinguishes a periodic function from other types of functions?
- It has a parabolic shape.
- It has a linear relationship between input and output values.
- It grows exponentially over time.
- It repeats its pattern over regular intervals. (correct)
The period of a periodic function is the shortest interval over which the function's pattern repeats.
The period of a periodic function is the shortest interval over which the function's pattern repeats.
True (A)
If a function $f(x)$ has a period of $k$, algebraically, how is this periodicity expressed?
If a function $f(x)$ has a period of $k$, algebraically, how is this periodicity expressed?
f(x + k) = f(x)
In a clock that is functioning improperly and moving twice as fast, the time between the minute hand pointing directly up at the 12 is an example of a shortened ______.
In a clock that is functioning improperly and moving twice as fast, the time between the minute hand pointing directly up at the 12 is an example of a shortened ______.
If angle $\theta$ is in standard position, what defines its reference angle?
If angle $\theta$ is in standard position, what defines its reference angle?
An angle in standard position and its coterminal angle always share the same terminal side.
An angle in standard position and its coterminal angle always share the same terminal side.
Convert 1 degree into radians.
Convert 1 degree into radians.
The measure of an angle in standard position is the ratio of the length of the arc of a circle centered at the origin created by the angle to the ______ of that same circle.
The measure of an angle in standard position is the ratio of the length of the arc of a circle centered at the origin created by the angle to the ______ of that same circle.
Which of the following statements about sine and cosine is correct?
Which of the following statements about sine and cosine is correct?
On the unit circle, the cosine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle.
On the unit circle, the cosine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle.
Given a point $(x, y)$ on the unit circle, what is the formula for $\tan(\theta)$?
Given a point $(x, y)$ on the unit circle, what is the formula for $\tan(\theta)$?
The sine function gives the ______ displacement from the x-axis.
The sine function gives the ______ displacement from the x-axis.
Match each trigonometric function with its reciprocal function:
Match each trigonometric function with its reciprocal function:
Which describes the graph of the function $f(x) = \sin(x)$?
Which describes the graph of the function $f(x) = \sin(x)$?
The graph of $f(x) = \cos(x)$ has rotational symmetry about the origin.
The graph of $f(x) = \cos(x)$ has rotational symmetry about the origin.
What is the term for half the difference between the maximum and minimum values of a sinusoidal function?
What is the term for half the difference between the maximum and minimum values of a sinusoidal function?
In a sinusoidal function, the ______ is the average of the maximum and minimum values.
In a sinusoidal function, the ______ is the average of the maximum and minimum values.
What does 'd' represent in the sinusoidal function $f(\theta) = \sin(\theta) + d$?
What does 'd' represent in the sinusoidal function $f(\theta) = \sin(\theta) + d$?
In the sinusoidal function $f(θ) = a \sin θ$, the constant a affects the period of the function.
In the sinusoidal function $f(θ) = a \sin θ$, the constant a affects the period of the function.
How are the period and frequency of a sinusoidal function related?
How are the period and frequency of a sinusoidal function related?
A horizontal translation is otherwise known as a ______ shift.
A horizontal translation is otherwise known as a ______ shift.
Match $y = 2\sin(\frac{1}{2}x)$ to a description.
Match $y = 2\sin(\frac{1}{2}x)$ to a description.
The transformations of functions $y = \cos(2x + \pi)$ consists of a horizontal dilation of SF = 2, and the phase shift to the right $PS = \frac{\pi}{2}$
The transformations of functions $y = \cos(2x + \pi)$ consists of a horizontal dilation of SF = 2, and the phase shift to the right $PS = \frac{\pi}{2}$
Which parameters of the sinusoidal function have to be calculated to determine $A\sin[B(x – C)] + D$?
Which parameters of the sinusoidal function have to be calculated to determine $A\sin[B(x – C)] + D$?
To model periodic data, ______ regression is used.
To model periodic data, ______ regression is used.
Which of the following defines the location of a point P on a unit circle that corresponds to an angle $\theta$ in standard position:
Which of the following defines the location of a point P on a unit circle that corresponds to an angle $\theta$ in standard position:
Coterminal angles must have the same cosine and sine values.
Coterminal angles must have the same cosine and sine values.
What does a negative angle in standard position indicate about the direction of rotation?
What does a negative angle in standard position indicate about the direction of rotation?
If $\tan(\theta) = 1$ and $0 \le \theta \le 2\pi$, then $\theta$ could be ______.
If $\tan(\theta) = 1$ and $0 \le \theta \le 2\pi$, then $\theta$ could be ______.
Which change occurs to a sinusoidal function if its period increases?
Which change occurs to a sinusoidal function if its period increases?
Increasing the amplitude of a sine function results in a graph that oscillates between higher and lower y-values, but the period remains unchanged.
Increasing the amplitude of a sine function results in a graph that oscillates between higher and lower y-values, but the period remains unchanged.
State the period and amplitude of the sine wave (y = 5 \sin(3x)).
State the period and amplitude of the sine wave (y = 5 \sin(3x)).
The general form of a sinusoidal function allows for horizontal shifts, which are also known as ______ shifts.
The general form of a sinusoidal function allows for horizontal shifts, which are also known as ______ shifts.
Where does the graph of y=cos x, -2$\pi$$\le$ x$\le$0 start?
Where does the graph of y=cos x, -2$\pi$$\le$ x$\le$0 start?
The midline of a sinusoidal function represents the average value between its maximum and minimum points.
The midline of a sinusoidal function represents the average value between its maximum and minimum points.
Describe the roles of parameters A and B in transforming $y = A\sin(Bx)$.
Describe the roles of parameters A and B in transforming $y = A\sin(Bx)$.
In the graphs of trigonometric functions, points of ______ indicate where a function's concavity changes.
In the graphs of trigonometric functions, points of ______ indicate where a function's concavity changes.
Which of the following describes a vertical dilation?
Which of the following describes a vertical dilation?
When modeling the distance of a point on a rotating Ferris wheel from the ground, a periodic function is appropriate because the motion repeats.
When modeling the distance of a point on a rotating Ferris wheel from the ground, a periodic function is appropriate because the motion repeats.
If (f(x) = A\sin(Bx + C) + D) models a sinusoidal wave, what parameter directly adjusts the Midline of the wave?
If (f(x) = A\sin(Bx + C) + D) models a sinusoidal wave, what parameter directly adjusts the Midline of the wave?
The period of the function represented by the equation f(t) = A sin(Bt) is ______.
The period of the function represented by the equation f(t) = A sin(Bt) is ______.
Flashcards
Periodic Relationship
Periodic Relationship
A relationship where output values repeat over equal input intervals.
Period
Period
The horizontal length of one complete cycle of a periodic function.
Angle
Angle
Two rays sharing a common vertex.
Initial Side
Initial Side
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Terminal Side
Terminal Side
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Positive Angle
Positive Angle
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Negative Angle
Negative Angle
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Standard Position (Angle)
Standard Position (Angle)
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Quadrantal Angle
Quadrantal Angle
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Radian Measure
Radian Measure
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Coterminal Angles
Coterminal Angles
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Reference Angle
Reference Angle
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Sine (sin)
Sine (sin)
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Cosine (cos)
Cosine (cos)
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Tangent (tan)
Tangent (tan)
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Sine Function (on unit circle)
Sine Function (on unit circle)
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Cosine Function (on unit circle)
Cosine Function (on unit circle)
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Period (sinusoidal)
Period (sinusoidal)
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Frequency
Frequency
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Amplitude
Amplitude
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Midline
Midline
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Vertical Translation
Vertical Translation
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Vertical Dilation
Vertical Dilation
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Horizontal Dilation
Horizontal Dilation
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Phase Shift
Phase Shift
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Study Notes
- Periodic phenomena are relationships between two variables resulting in output values taking on and repeating a pattern over equal input intervals.
- The graph of a periodic relationship is determined by a single cycle.
- The period is the length of the input interval for one cycle.
- Algebraically, a periodic function is written as f(x + k) = f(x) for all k, where k is the function's period.
Sine, Cosine, and Tangent
- An angle has two rays with a common vertex.
- The initial side is a ray on the positive x-axis.
- The terminal side is a ray that rotates around the origin.
Positive and Negative Angles
- A positive angle is a rotating counter- clockwise
- A negative angle is rotating clockwise
Standard Position of an Angle
- An angle whose vertex is at the origin.
- An angle whose initial side is the positive x-axis.
Quadrantal Angle
- An angle is a standard position whose terminal side lies on an axis.
- The radian measure of an angle in standard position is the ratio of the length of the arc of a circle centered at the origin created by the angle to the radius of that same circle.
- One revolution is 2π radians, equivalent to 360°.
- 1 degree = π/180 radians.
- 1 radian = 180/π ≈ 57 degrees.
Arc Length
- Arc Length indicated is s, and Radius indicated is r
- Θ = central angle
- r = radius
- s = arc length
- Formula for arc length: Θ= s/r
Converting between Degrees and Radians:
- To convert from degrees to radians, multiply by π/180.
- To convert from radians to degrees, multiply by 180/π.
- Coterminal angles have the same initial and terminal sides but have different measures.
- Two methods exist
- Degrees: Θ ± 360°k, where k equals any integer
- Radians: Θ ± 2πk, where k equals any integer
Reference Angle
- A reference angle, in standard position, is the positive, acute angle formed by the terminal side of [Θ] and the x-axis
Reference Angle by Quadrant:
- Quadrant II: [Θ’ = π - Θ]
- Quadrant I: [Θ’ = Θ]
- Quadrant III: [Θ’ = Θ - π]
- Quadrant IV: [Θ’ = 2π - Θ]
SOH CAH TOA
- Sine [Θ]= opposite / hypotenuse = y/r
- Cosine [Θ]= adjacent / hypotenuse = x/r
- Tangent [Θ]= opposite / adjacent = y/x
- Also known as tan [Θ]= sin [Θ]/ cos [Θ]
Quadrantal Angles
- An angle In standard position equals terminal side coincident with the x or y axis
Cosine of Angle
- Ratio of the horizontal displacement P from y-axis Unit circle cosine equals x coordinate of point P
Tangent of Angle
- Exists given a slope of the terminal ray
- Tangent equals the ratio of the vertical displacement over horizontal displacement
Unit Circle
- [Sin Θ = y], [Csc Θ= 1/y]
- [Cos Θ = x], [Sec Θ= 1/x]
- [Tan Θ = y/x], [Cot Θ= x/y]
- Range of Sine: [-1,1]
- Range of Cosine: [-1,1]
Unit Circle special cases
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Sine and Cosine of angles are multiples of π/4.
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An angle in standard position in a unit circle with a radian measure of π/4 is an angle in an isosceles right triangle.
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In a unit circle, the coordinates of Point P for an angle in standard position with a measure of π/4 in Quadrant 1, the coordinates are (√2/2, √2/2).
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Sine and Cosine of angles that are multiples of π/3
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An angle in standard position in a unit circle with a radian measure of π/3 is an angle in an equilateral triangle.
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In a unit circle, the coordinates of Point P for an angle in standard position with a measure of π/3 in Quadrant 1, the coordinates are (1/2,√3/2).
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The sine function of an angle in standard position on the unit circle gives vertical displacement from the x-axis.
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The cosine function of an angle in standard position on the unit circle gives horizontal displacement from the y-axis.
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Cosine is just a translation of Sine
Sinusoidal Characteristics
- The period of a sinusoidal function means the smallest positive value k such that f(Θ+k) = f(Θ) for all (Θ) in the domain.
- The frequency means the horizontal dilation or # of times the pattern repeats
- The amplitude is 1/2 the difference between the max and min. value in 1 period
- The midline is the average of the maximum value and the minimum value.
Transforming sinusoidal
- [f(x)=Asin[B(x−C)]+D]
- Where. Reflection happens on x-axis, vertical shift, vertical dilation and horizontal shift are present
- Vertical Translation: f(Θ) = sinΘ + d or f(Θ) = cosΘ + d, a vertical shift of the graph of sine or cosine, including the midline, by d units.
- Vertical Dilation: f(Θ) = a sinΘ or f(Θ) = a cosΘ. A vertical dilation of the graph of sine or cosine, where amplitude changes by a factor of |a|. also changed by a factor of.a..
- Horizontal Dilation: [f(Θ)=sin(bΘ)] or [f(Θ)=cos(bΘ)]. horizontal dilation of the graph of sine or cosine, where period also changes by a factor of
- Additive Transformation (Horizontal Shift): [f(Θ)=sin(Θ+c)] or [f(Θ)=cos(Θ+c)]. A horizontal translation, also known as the phase shift shift is present of the graph of sine or cosine by c units
Steps for fitting data to the Sinusoidal Function:
$$f(x) = A \sin[B(x – C] + D$$
- Determine A or Amplitude. A = Largest Data Value – Smallest Data Value/2
- Determine D or Vertical Translation. Vertical Translation equals a midline D = Largest Data Value + Smallest Data Value / 2
- Determine B (Horizontal Stretch) by first finding P the Period or the time if takes for data to repeat. So: P = [2π/B], [B =2π/P]
Determine C or the Phase Shift (Horizontal Shift), by using the ordered pair (x, (f(x)) from data where f(x) equals the smallest, answers depend on ordered pair selected so consistency is maintained [f(x)=Asin[B(x−C)] + D], Make sure you factor
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