Section 4.1
Change in Period: Graphing 𝑦 = sin 𝒃𝑥 or 𝑦 = cos 𝒃𝑥
Graph 𝒚 = 𝐬𝐢𝐧 𝟐𝒙 over a two-period interval.
Amplitude:
𝑥
(angle in radians)
Period:
y
0
𝜋
4
𝜋
2
3𝜋
4
𝜋
5𝜋
4
3𝜋
2
7𝜋
4
2𝜋
The coefficient “b” represents the number of times sine will complete a cycle between 0 to 2𝜋.
If we change the coefficient of x that will change the period of the function.
The input angle now is “bx” will still cycle through all angle values from 0 to 2𝜋 or 0 ≤ 𝑏𝑥 ≤ 2𝜋.
So in order for the function to complete a cycle, the value of x must be 0 ≤ 𝑥 ≤
The period of any sine or cosine function is
𝟐𝝅
𝒃
2𝜋
𝑏
.
Changes made to the inside of the function affect the graph horizontally. In other words, it changes the
x-values used for the graph.
To find the x-values that will be used for 1 period,
Set up the inequality 𝟎 ≤ 𝒊𝒏𝒔𝒊𝒅𝒆 𝒐𝒇 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 ≤ 𝟐𝝅 then solve for x. This gives you the start and end points
for one period.
Next, find the midpoints =
𝑠𝑡𝑎𝑟𝑡+𝑒𝑛𝑑
2
.
Basic steps to graph one period of Sine and Cosine.
1) Find the x-values. Set up the inequality 𝟎 ≤ 𝒊𝒏𝒔𝒊𝒅𝒆 𝒐𝒇 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 ≤ 𝟐𝝅 then solve for x.
This gives you the start and end points for one period. Find the midpoints =
𝑠𝑡𝑎𝑟𝑡+𝑒𝑛𝑑
2
.
2) Draw a dotted horizontal line for the middle of the graph. (Vertical shift will determine the middle)
3) Draw in the basic shape for sine or cosine using the amplitude to determine the vertical distance from the
middle of the graph.
𝟏
Example 1: Graph 𝒚 = 𝐜𝐨𝐬 𝟐 𝒙 over a one-period interval.
Amplitude:
Period:
Example 2: Graph 𝒚 = 𝐜𝐨𝐬 𝟒𝒙 over a one-period interval.
Amplitude:
Period:
𝟏
Example 3: Graph 𝒚 = 𝐬𝐢𝐧 𝟑 𝒙 over a one-period interval.
Amplitude:
Period: