1. Think about riding a bike and pumping the pedals at a constant rate of one revolution each second.
How does the graph of the height of one of your feet compare with the graph of a sine function?

2. 13-7 Translating Trigonometric Functions
Today’s Objective:
I can write and graph a trigonometric functions.

3. Translating Functions
Vertical
Horizontal
𝑓(𝑥)
𝑓 𝑥 +𝑘
𝑓(𝑥)
𝑓(𝑥−ℎ)
Translate h units horizontally
Translate k units vertically
Phase Shift
Midline: y = k
𝒚= 𝐬𝐢𝐧 𝒙− 𝝅 𝟐
𝒚= 𝐜𝐨𝐬 𝒙 +𝟐
= 𝝅 𝟐 h
𝒚=𝟐
𝒚= 𝐬𝐢𝐧 (𝒙)
𝒚= 𝐜𝐨𝐬 (𝒙)

4. Family of Trigonometric Functions
Parent Functions
Transformed Function
𝑦= sin 𝑥
𝑦= 𝒂 sin 𝒃(𝑥−𝒉) +𝒌
𝑦= cos 𝑥
𝑦= 𝒂 cos 𝒃(𝑥−𝒉) +𝒌
𝑎 =
Amplitude: Vertical stretch or shrink
2𝜋 𝑏 =
Period ℎ=
Phase shift: Horizontal shift 𝑘=
Vertical shift : y = k is midline

5. Graph each function on interval from 0 to 2π
Amplitude: 1
𝑦= sin 𝑥+ 𝜋 2 −2
Midline:
𝑦=−2
Period: 2𝜋
Phase Shift:
Left 𝜋 2
Graphing:
Sketch in Midline (y = k)
Graph beginning point with phase shift.
Graph remaining four points.

6. Graph each function on interval from 0 to 2π
𝑦=2 cos 𝑥− 𝜋
Amplitude: 2
Midline:
𝑦=1
Period: 2𝜋
Phase Shift:
Right 𝜋 3
Graphing:
Sketch in Midline (y = k)
Graph beginning point with phase shift.
Graph remaining four points.

7. Write a sine and cosine function for the graph.
𝝅 𝟒
𝑦= sin (𝑥− ) + 𝟑 𝟏 𝟏
𝑦= 𝒂 cos 𝒃(𝑥−𝒉) +𝒌
𝟑𝝅 𝟒
𝑦= cos (𝑥− ) + 𝟑 𝟏 𝟏
W.S. Translating Sine and Cosine Functions
Ch. Test Review p. 897: 1, 3-14, 17-20, 25-30, 32
p. 880: 22-25, 27, 28, 31, 33, 44, 45

8. Graph each function on interval from 0 to 2π
Amplitude: 3
𝑦=3 sin 2 𝑥− 𝜋
Midline:
𝑦=2
Period: 𝜋
Phase Shift:
Right 𝜋 6
Graphing:
Sketch in Midline (y = k)
Graph beginning point with phase shift.
Graph remaining four points.