[Func-Num] 8.6 Graphs of Sin and Cos

📋 This is my note-taking from what I learned in the class “Math175-002 Functions & Number Systems”

Graphs of “y = a sin(bx + c)” and “y = a cos(bx + c)”

1. Amplitude

   y_max = |a|

   y_min = -|a|

2. Period → P = 2π/b

   P/4 = step on x-axis (five x values needed on x-axis to have graph for one period)

3. Shift or Displacement = -c/b

   -c/b = Starting x value (c is called phase angle)

Note: You must have your Calculator on Radians

Example

Example 1: Sketching graph of “y = a sin(bx + c)”

Sketch the graph y = 2 sin(3x - π).

Solution
a = 2	Amplitude = ymax = |a| = 2, ymin = -|a| = -2
b = 3	Period(cycle) = P = 2π/b = 2π/3, P/4 = 2π/12 = π/6 = Step on x-axis
c = -π	Shift(Displacement) = -c/b = -(-π)/3 = π/3 = Starting x value on x-axis
x starts at π/3 and have five values on x-axis at a step = π/6

Sample Calculation: Calculator on Radians

• x = π/3 → y = 2 sin(3(π/3)-π) = 0

x	\({π} \over {3}\)	\({π} \over {2}\)	\({2π} \over {3}\)	\({5π} \over {6}\)	π
y	0	2	0	-2	0

Example 2: Sketching graph of “y = a cos(bx + c)”

Sketch the graph y = -cos(2x + π/6).

Solution
a = -1	Amplitude = ymax = |a| = 1, ymin = -|a| = -1
b = 2	Period(cycle) = P = 2π/b = 2π/2 = π, P/4 = π/4 = Step on x-axis
c = π/6	Shift(Displacement) = -c/b = -(π/6)/2 = -π/12 = Starting x value on x-axis
x starts at -π/12 and have five values on x-axis at a step = π/4

Sample Calculation: Calculator on Radians

• x = -π/12 → y = -cos(2(-π/12) + π/6) = -1

x	\({-π} \over {12}\)	\({π} \over {6}\)	\({5π} \over {12}\)	\({2π} \over {3}\)	\({11π} \over {12}\)
y	-1	0	1	0	-1

Exercise

Exercise 1

For the function below:

• a) What is amplitude , y_max and y_min.
• b) What is the period?
• c) What is the shift (horizontal)?
• d) Graph one period and show values at 1/4th of the period, the shift and the amplitude.

y = sin(x - π/6)

Exercise 2

For the function below:

• a) What is amplitude , y_max and y_min.
• b) What is the period?
• c) What is the shift (horizontal)?
• d) Graph one period and show values at 1/4th of the period, the shift and the amplitude.

y = -cos(2x - π)

Exercise 3

For the function below:

• a) What is amplitude , y_max and y_min.
• b) What is the period?
• c) What is the shift (horizontal)?
• d) Graph one period and show values at 1/4th of the period, the shift and the amplitude.

Exercise 4

For the function below:

• a) What is amplitude , y_max and y_min.
• b) What is the period?
• c) What is the shift (horizontal)?
• d) Graph one period and show values at 1/4th of the period, the shift and the amplitude.

Exercise 5

For the function below:

• a) What is amplitude , y_max and y_min.
• b) What is the period?
• c) What is the shift (horizontal)?
• d) Graph one period and show values at 1/4th of the period, the shift and the amplitude.

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