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Graphing Sine and Cosine Functions: Steps and Examples, Lecture notes of Calculus

The steps to graph sine and cosine functions, including identifying the amplitude, period, phase shift, and vertical shift, and provides examples for both sine and cosine functions. Follow the steps to determine the endpoints, middle point, and critical points, then connect the points with a smooth curve.

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Steps for graphing Sine and Cosine Functions:
Step 1: Make sure the equation is written as 𝑦=asin(𝑏𝑏𝑐)+𝑑 or 𝑦=asin 𝑏�𝑏 𝑐
𝑏+𝑑.
Step 2: Identify the following:
a) Amplitude
b) Period
c) Phase Shift
d) Vertical Shift
Step 3: Draw your resting line of y = d
Step 4: Determine your initial endpoint of the first period �−𝑐
𝑏
a) Positive Sine starts at a Resting position and goes to a Maximum
b) Negative Sine starts at a Resting position and goes to a Minimum
c) Positive Cosine starts at a Maximum
d) Negative Cosine starts at a Minimum
Step 5: Determine your last endpoint of the first period �−𝑐
𝑏+2𝜋
𝑏.
Step 6: Determine your middle point of the first period (half the distance horizontally between
your two endpoints).
Step 7: Determine your other two critical points (half the distance horizontally between the
middle point and each end point)
Step 8: Connect your points using a SMOOTH curve.
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Steps for graphing Sine and Cosine Functions:

Step 1: Make sure the equation is written as 𝑦 = asin(𝑏𝑏 − 𝑐) + 𝑑 or 𝑦 = asin 𝑏 �𝑏 − 𝑐 𝑏 �^ +^ 𝑑. Step 2: Identify the following: a) Amplitude b) Period c) Phase Shift d) Vertical Shift Step 3: Draw your resting line of y = d

Step 4: Determine your initial endpoint of the first period �− 𝑐 𝑏 � a) Positive Sine starts at a Resting position and goes to a Maximum b) Negative Sine starts at a Resting position and goes to a Minimum c) Positive Cosine starts at a Maximum d) Negative Cosine starts at a Minimum

Step 5: Determine your last endpoint of the first period �− 𝑐 𝑏 +^

2𝜋 𝑏 �. Step 6: Determine your middle point of the first period (half the distance horizontally between your two endpoints). Step 7: Determine your other two critical points (half the distance horizontally between the middle point and each end point) Step 8: Connect your points using a SMOOTH curve.

Example 1: 𝑦 − 3 = −4 cos � 𝜃 2 +^

𝜋 4 �

Step 1: Make sure the equation is written as 𝑦 = asin(𝑏𝑏 − 𝑐) + 𝑑 or 𝑦 = asin 𝑏 �𝑏 − 𝑐 𝑏 �^ +^ 𝑑. 𝑦 = −4 cos � 𝜃 2 +^

𝜋 4 �^ + 3 Step 2: Identify the following: e) Amplitude: a = -4, amplitude = 4 f) Period: b = 1 2 , period =^

2𝜋 1 2

g) Phase Shift: 𝑐 = 𝜋 4 , b =^

1 2 , phase shift =^ − �

𝜋 4 1 2

𝜋 2 h) Vertical Shift: d = 3, vertical shift = 3 Step 3: Draw your resting line of y = d

Step 4: Determine your initial endpoint of the first period �− 𝑐 𝑏 �^ =^ −^

𝜋 2 =^ 𝑏 e) Positive Sine starts at a Resting position and goes to a Maximum f) Negative Sine starts at a Resting position and goes to a Minimum g) Positive Cosine starts at a Maximum h) Negative Cosine starts at a Minimum