Introduction to Sine Graphs

Warm-up (2:30 m)• For the graph below, identify the max, min, y-

int, x-int(s), domain and range.

Fill in the table below. Then use the points to sketch the graph of y = sin t

t 0

sin t2

π4

π4

π3

π 2π2π3

4π5

4π7

4π

2π

4π3 π 2π4

π52

π34

π7

Reflection Questions3. What is the max of y = sin t? What is the min?

4. What is the y-int? What are the x-intercepts?

5. What is the domain? What is the range?

Reflection Questions, cont.6. What do you think would happen if you

extended the graph beyond 2π?

7. How would extending the graph affect the domain and the x-intercepts?

Periodicity• Trigonometric graphs are

periodic because the pattern of the graph repeats itself

• How long it takes the graph to complete one full wave is called the period

0

2

–21 Period 1 Period

Period: π

π 2π

Periodicity, cont.

2tsin)t(f )t4sin()t(f

2 2

–2 –2

–2π2π –π

π

Your Turn:• Complete problems 1 – 3 in the guided notes.

Maximum

Minimum

Domain

Range

Period

Maximum

Minimum

Domain

Range

Period

Maximum

Minimum

Domain

Range

Period

1. f(t) = –3sin(t) 2.

3. f(t) = sin(5t)

4tsin2)t(f

Calculating Periodicity• If f(t) = sin(bt), then period =• Period is always positive

4. f(t) = sin(–6t) 5.

6.

|b|π2

4tsin)t(f

4t3sin)t(f

Your Turn:• Calculate the period of the following graphs:

7. f(t) = sin(3t) 8. f(t) = sin(–4t)

9. 10. f(t) = 4sin(2t)

11. 12.

5

t2sin6)t(f

8

tsin4)t(f

4tsin)t(f

Amplitude• Amplitude is a trigonometric graph’s greatest distance

from the middle line. (The amplitude is half the height.)• Amplitude is always positive.

– If f(t) = a sin(t), then amplitude = | a |

2)tsin(21)t(f

f(t) = 3sin(t) + 1

Calculating Amplitude Examples17. f(t) = 6sin(4t) 18. f(t) = –5sin(6t)

19. 20.)tsin(32)t(f

3tsin

51)t(f

Your Turn:• Complete problems 21 – 26 in the guided

notes

21. f(t) = –2sin(t) + 1 22. f(t) = sin(2t) + 4

23. f(t) = sin(2t) 24. f(t) = –3sin(t)

25. 26.

3tsin3.0)t(f )t3sin(

21)t(f

Sketching Sine Graphs – Single Smooth Line!!!