Advanced Math
Sine Graphs & Transformations
Trig Graphs D1
Name:
Graph a unit circle (in degrees) and label all the points.
Using the axis below, graph one cycle of the sine function.
Where does the cycle end? __________
That is called the ______________________________
What is the maximum value for sine? __________
What is the minimum value for sine? __________
Define: amplitude: _________________________________________________________________
_________________________________________________________________
Define: phase shift: ________________________________________________________________
Define: vertical shift: _______________________________________________________________
Define: reflection: __________________________________________________________________
The general form for a sine equation is:
Where ________ is the amplitude, which is also a vertical _________________ or ________________
If 𝒂 is negative there is a ____________________ over the _____________
ℎ is the
𝑘 is the
1
Example 1: For each function, state the amplitude, if there is a reflection, the phase shift and
the vertical shift. Write “none” for transformations that do not exist.
1
1. 𝑦 = 2 sin 𝜃
2. 𝑦 =
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
4. 𝑦 = − 2sin 𝜃
5. 𝑦 = − sin 𝜃
6. 𝑦 = sin 𝜃 − 3
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
7. 𝑦 = sin 𝜃 + 4
8. 𝑦 = sin 𝜃 − 2
9. 𝑦 = sin(𝜃 − 90°)
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
10. 𝑦 = sin(𝜃 + 180°) − 2
11. 𝑦 = 2sin(𝜃 + 180°)
12. 𝑦 = 3sin(𝜃 + 90°)
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
2
sin 𝜃
1
2
3. 𝑦 = − sin 𝜃
2
Example 2: Graph the functions from Example 1.
Step #1: Start by labeling the 𝒙-axis in degrees and the 𝒚-axis in integer units. Then graph the
parent function 𝒚 = 𝐬𝐢𝐧 𝜽.
Step #2: List the transformations
Step #3: Graph each transformation – one at a time
1.
𝑦 = 2 sin 𝜃
Transformations:
3.
𝑦 = − sin 𝜃
Transformations:
2.
𝑦=
1
2
sin 𝜃
Transformations:
4.
𝑦 = − 2sin 𝜃
Transformations:
3
5.
1
𝑦 = − 2 sin 𝜃
Transformations:
7.
𝑦 = sin 𝜃 + 4
Transformations:
6.
𝑦 = sin 𝜃 − 3
Transformations:
8.
𝑦 = sin 𝜃 − 2
Transformations:
4
9.
𝑦 = sin(𝜃 − 90°)
Transformations:
11.
𝑦 = 2sin(𝜃 + 180°)
Transformations:
10.
𝑦 = sin(𝜃 + 180°) − 2
Transformations:
12.
𝑦 = 3sin(𝜃 + 90°)
Transformations:
5
Advanced Math
Cosine Graphs & Transformations
Trig Graphs D2
Name:
Graph a unit circle (in degrees) and label all the points.
Using the axis below, graph one cycle of the cosine function.
Where does the cycle end? __________
That is called the ______________________________
What is the maximum value for cosine? __________
What is the minimum value for cosine? __________
Define: amplitude: _________________________________________________________________
_________________________________________________________________
Define: phase shift: ________________________________________________________________
Define: vertical shift: _______________________________________________________________
Define: reflection: __________________________________________________________________
The general form for a cosine equation is:
Where ________ is the amplitude, which is also a vertical _________________ or ________________
If 𝒂 is negative there is a ____________________ over the _____________
ℎ is the
𝑘 is the
6
Example 1: For each function, state the amplitude, if there is a reflection, the phase shift and
the vertical shift. Write “none” for transformations that do not exist.
1
1. 𝑦 = 3 cos 𝜃
2. 𝑦 =
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
4. 𝑦 = − 2cos 𝜃
5. 𝑦 = − cos 𝜃
6. 𝑦 = cos 𝜃 − 2
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
7. 𝑦 = cos 𝜃 + 3
8. 𝑦 = cos(𝜃 − 180°)
9. 𝑦 = cos(𝜃 + 90°) + 1
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
10. 𝑦 = cos(𝜃 + 180°) − 2
11. 𝑦 = 2 cos(𝜃 − 90°)
12. 𝑦 = 3cos(𝜃 + 180°)
Amplitude:_______
Amplitude:_______
Amplitude:_______
Reflection:_______
Reflection:_______
Reflection:_______
Phase Shift:__________
Phase Shift:__________
Phase Shift:__________
Vertical Shift:__________
Vertical Shift:__________
Vertical Shift:__________
2
cos 𝜃
1
2
3. 𝑦 = − cos 𝜃
7
Example 2: Graph the functions from Example 1.
Step #1: Start by labeling the 𝒙-axis in degrees and the 𝒚-axis in integer units. Then graph the
parent function 𝒚 = 𝐜𝐨𝐬 𝜽.
Step #2: List the transformations
Step #3: Graph each transformation – one at a time
1.
𝑦 = 3 cos 𝜃
Transformations:
3.
𝑦 = − cos 𝜃
Transformations:
2.
𝑦=
1
2
cos 𝜃
Transformations:
4.
𝑦 = − 2cos 𝜃
Transformations:
8
5.
1
𝑦 = − 2 cos 𝜃
Transformations:
7.
𝑦 = cos 𝜃 + 3
Transformations:
6.
𝑦 = cos 𝜃 − 2
Transformations:
8.
𝑦 = cos(𝜃 − 180°)
Transformations:
9
9.
𝑦 = cos(𝜃 + 90°) + 1
Transformations:
11.
𝑦 = 2 cos(𝜃 − 90°)
Transformations:
10.
𝑦 = cos(𝜃 + 180°) − 2
Transformations:
12.
𝑦 = 3cos(𝜃 + 180°)
Transformations:
10