Graphing Trigonometric Functions - Worksheet 15 Key
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Questions and Answers

What is the period of the function y = 3 sin(x)?

2Ï€

What is the amplitude of the function y = 3 sin(x)?

3

What is the phase shift of the function y = 3 sin(x)?

0

What is the vertical shift of the function y = 3 sin(x)?

<p>0</p> Signup and view all the answers

What is the period of the function y = -2cos(x)?

<p>2Ï€</p> Signup and view all the answers

What is the period of the function y = cos(x - π/2)?

<p>2Ï€</p> Signup and view all the answers

What is the amplitude of the function y = cos(x - π/2)?

<p>1</p> Signup and view all the answers

What is the phase shift of the function y = cos(x - π/2)?

<p>Ï€/2</p> Signup and view all the answers

What is the vertical shift of the function y = cos(x - π/2)?

<p>0</p> Signup and view all the answers

What is the period of the function y = -sin(x + π/3)?

<p>2Ï€</p> Signup and view all the answers

What is the amplitude of the function y = -sin(x + π/3)?

<p>1</p> Signup and view all the answers

What is the phase shift of the function y = -sin(x + π/3)?

<p>-Ï€/3</p> Signup and view all the answers

What is the vertical shift of the function y = -sin(x + π/3)?

<p>0</p> Signup and view all the answers

What is the period of the function y = (1/3)cos(x/2 + π/3)?

<p>4Ï€</p> Signup and view all the answers

What is the amplitude of the function y = (1/3)cos(x/2 + π/3)?

<p>1/3</p> Signup and view all the answers

What is the phase shift of the function y = (1/3)cos(x/2 + π/3)

<p>-2Ï€/3</p> Signup and view all the answers

What is the vertical shift of the function y = (1/3)cos(x/2 + π/3)?

<p>0</p> Signup and view all the answers

What is the period of the function y = cos(3x - 2Ï€) + 4?

<p>2Ï€/3</p> Signup and view all the answers

What is the amplitude of the function y = cos(3x - 2Ï€) + 4?

<p>1</p> Signup and view all the answers

What is the phase shift of the function y = cos(3x - 2Ï€) + 4?

<p>2Ï€/3</p> Signup and view all the answers

What is the vertical shift of the function y = cos(3x - 2Ï€) + 4?

<p>4</p> Signup and view all the answers

What is the period of the function y = 2tan(x/4) - 3?

<p>4Ï€</p> Signup and view all the answers

What is the period of the function y = (1/3)tan(-2x - π) + 1 ?

<p>Ï€/2</p> Signup and view all the answers

What is the period of the function y = csc(x - π/4) - 2?

<p>2Ï€</p> Signup and view all the answers

What is the period of the function y = (1/3)cot(2x + 3Ï€/2) + 1?

<p>Ï€/2</p> Signup and view all the answers

Flashcards

Period of a trigonometric function

The distance between consecutive peaks or troughs on a graph of a periodic function.

Amplitude of a trigonometric function

The maximum displacement of a periodic function from its equilibrium position. It is half the difference between the maximum and minimum values.

Phase Shift of a trigonometric function

The horizontal shift of a periodic function from its standard position.

Vertical Shift of a trigonometric function

The vertical shift of a periodic function from its standard position.

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Period of sin(x) and cos(x)

The period of the function y=sin(x) or y=cos(x), or any transformation of these functions, is 2Ï€.

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Amplitude of sin(x) and cos(x)

The amplitude of the function y=sin(x) or y=cos(x) is always 1.

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Phase Shift of sin(x) and cos(x)

The phase shift of the function y=sin(x) or y=cos(x) is always 0.

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Vertical Shift of sin(x) and cos(x)

The vertical shift of the function y=sin(x) or y=cos(x) is always 0.

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Period of tan(x)

The period of the function y=tan(x) is π.

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Period of sec(x) and csc(x)

The period of the function y=sec(x) or y=csc(x) is 2Ï€, just like sine and cosine. The period is found the same way.

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Period of cot(x)

The period of the function y=cot(x) is π, just like tangent. The period is found the same way.

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General form of sine and cosine functions

The graph of y=a sin(bx+c) +d, or of y=a cos(bx+c) +d, has the following properties:

Period: 2π/b Amplitude: |a| Phase shift: −c/b Vertical shift: d

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Reflection across the x-axis in trigonometric functions

When a function is multiplied by a negative sign, it is reflected across the x-axis.

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Period change in trigonometric functions

For the function y=sin(bx) or y=cos(bx), the period is 2Ï€/b. A multiplier on the x term inside the function changes the period.

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Amplitude change in trigonometric functions

For the function y = a sin(x) or y = a cos(x), the amplitude is |a|. A multiplier on the function itself changes the amplitude.

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Phase shift change in trigonometric functions

For the function y = sin(x + c) or y = cos(x + c), the phase shift is −c. A constant added to the x term inside the function changes the phase shift.

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Vertical shift change in trigonometric functions

For the function y = sin(x) + d or y = cos(x) + d, the vertical shift is d. A constant term added to the function itself changes the vertical shift.

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General form of tangent function

The graph of y=tan(bx +c) + d has the following properties:

Period: π/b Phase shift: -c/b Vertical shift: d

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General form of secant and cosecant functions

The graph of y=sec(bx +c) + d or y=csc(bx+c)+d, has the following properties:

Period: 2π/b Phase shift: −c/b Vertical shift: d

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General form of cotangent function

The graph of y=cot(bx+c) +d has the following properties:

Period: π/b Phase shift: -c/b Vertical shift: d

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Even and odd functions in trigonometry

The function y = sin(−x) is equivalent to y = − sin(x). The function y = cos(−x) is equivalent to y = cos(x). These are examples of odd and even functions, respectively.

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Even and odd functions in trigonometry

The function y=tan(−x) is equivalent to y = −tan(x).

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Periodic function

A function is periodic if its graph repeats over equal intervals.

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Sine and cosine ratios

The sine function is the ratio of the opposite side to the hypotenuse in a right triangle, while the cosine function is the ratio of the adjacent side to the hypotenuse.

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Tangent ratio

The tangent function is the ratio of the opposite side to the adjacent side in a right triangle.

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Secant function

The secant function is the reciprocal of the cosine function.

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Cosecant function

The cosecant function is the reciprocal of the sine function.

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Cotangent function

The cotangent function is the reciprocal of the tangent function.

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Graphs of secant and cosecant

The graph of y=sec(x) is found by taking the reciprocal of the graph of y=cos(x). The graph of y=csc(x) is found by taking the reciprocal of the graph of y=sin(x).

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Graph of cotangent

The graph of y=cot(x) is found by taking the reciprocal of the graph of y=tan(x).

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Study Notes

Graphing Trigonometric Functions

  • Worksheet 15 Key provides examples of graphing trigonometric functions with different transformations.
  • Each example demonstrates a specific trigonometric function (sine, cosine, tangent, secant, cosecant, cotangent) with variations in amplitude, period, phase shift, and vertical shift.
  • The transformations affect the graph's shape and position.

Function Examples

  • y = 3sin(x):

    • Period: 2Ï€
    • Amplitude: 3
    • Phase Shift: 0
    • Vertical Shift: 0

  • y = sin(3x):

    • Period: 2Ï€/3
    • Amplitude: 1
    • Phase Shift: 0
    • Vertical Shift: 0

  • y = -2cos(x):

    • Period: 2Ï€
    • Amplitude: 2
    • Phase Shift: 0
    • Vertical Shift: 0

  • y = cos(x - Ï€/2):

    • Period: 2Ï€
    • Amplitude: 1
    • Phase Shift: Ï€/2
    • Vertical Shift: 0

  • y = -sin(x) + (x + Ï€/3):

    • Period: 2Ï€
    • Amplitude: 1
    • Phase Shift: -Ï€/3
    • Vertical Shift: 0

  • y = sin(2x - Ï€):

    • Period: Ï€
    • Amplitude: 1
    • Phase Shift: Ï€/2
    • Vertical Shift: 0

  • y = cos(x/3):

    • Period: 8Ï€
    • Amplitude: 1
    • Phase Shift: 0
    • Vertical Shift: 0

  • y = cos(3x − 2Ï€) + 4:

    • Period: 2Ï€/3
    • Amplitude: 1
    • Phase Shift: 2Ï€/3
    • Vertical Shift: 4

  • y = sin(x + 1) - 2 -Amplitude =1 -Period =2Ï€ -Phase Shift = -1

    • Vertical Shift=-2

  • Additional examples include variations of tangent, cotangent, secant, and cosecant functions with similar transformations.

Graphing Strategy

  • Graphing strategies mention showing several cycles to illustrate phase shifts; finding parameters like period and amplitude by referring to the base function.

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Related Documents

Worksheet 15 KEY - Graphing Trigonometric Functions PDF

Description

This quiz focuses on the graphing of trigonometric functions, specifically through Worksheet 15 Key. Each example illustrates how various transformations, such as amplitude, period, and phase shift, impact the shape and position of sine, cosine, tangent, and other functions. Test your understanding of these concepts with detailed function examples.

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