Graphs of Trigonometric Functions

Graphs of Trigonometric Functions

By Elo Cheng

Graph of the Sine Function

​The sine function is a set of ordered pairs of real numbers. Each ordered pair can be represented as a point of the coordinate plane. The domain of the sine function is the set of real numbers, that is, every real number is a first element of one pair of the function.

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Multiple Choice

What is the domain and range of y=sinx?

• D: (-∞, ∞)
  R: [-1, 1]
  
• D: (-∞, ∞)
  R: [-π, π]
• D: (∞, -∞)
  R: [-1, 1]
  
• D: (-∞, ∞)
  R: [-∞, ∞]
  

On this graph, we plot the points whose coordinates are given in the table. Through these points, we draw a smooth curve.

Each time we increase or decrease the value of x-coordinates by a multiple of 2π, the basic curve is repeated. Each portion of the graph in an interval of 2π is one cycle of the sine curve.

The graph of the function y=sinx is called a periodic function with a period of 2π.

Camparing the graph of the unit circle and the graph of y=sinx in the figures above.

Multiple Choice

The period of a function is 

• how long one cycle is before the function repeats
• how high and low the function goes from the center line
• the range of the function

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Multiple Choice

What does a negative 'a' value cause?

• A negative amplitude

• A reflection across the y-axis

• No effect

• A reflection across x-axis

Multiple Choice

The __________ represents half the distance between the max and min of a sine or cosine graph. 

• Period
• Frequency
• Extreme Values
• Amplitude

Multiple Choice

What is the amplitude of the graph? 

• 1
• 3
• 6
• 9

Multiple Choice

The amplitude of a sine or cosine wave measures the distance from the midline to the "top" of the wave, which is also the distance from the midline to the "bottom" of the wave.

• True

• False

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Multiple Choice

Calculate the period of the curve

• $$4\pi$$  

• $$2\pi$$  

• 2

• 0

Multiple Choice

Calculate the period of the curve

• $$4\pi$$

• $$2\pi$$

• 1

• -1

Multiple Choice

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

• π
• 2π
• 3π
• 4π

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Multiple Choice

In the equation

$$f\left(\theta\right)=7\sin\left(\theta\right)+2$$  ,  what are the minimum and maximum values of the function?

• min = 5; max = 9

• min = 2; max = 7
  

• min = -5; max = 9

• min = -9; max = -5