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 sin t π 2π

π 2π

Values of Sine t sin(t) π 2π

Trigonometric Graphs Repeat!!! Maximum: Minimum: Range: Domain:

Reflection Questions What is the max of y = sin t? What is the min? What is the y-int? What are the x-intercepts? What is the domain? What is the range?

Reflection Questions, cont. What do you think would happen if you extended the graph beyond 2π? How would extending the graph affect the domain and the x-intercepts?

Periodicity Period: π 2 π 2π 1 Period 1 Period –2 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 π 2π 1 Period 1 Period –2

Periodicity, cont. 2 2 –2π π 2π –π –2 –2

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

Maximum Minimum Domain Range Period Maximum Minimum Domain Range 1. f(t) = –3sin(t) 2. Maximum Minimum Domain Range Period Maximum Minimum Domain Range Period 3. f(t) = sin(5t) Maximum Minimum Domain Range Period

Calculating Periodicity If f(t) = sin(bt), then period = Period is always positive 4. f(t) = sin(–6t) 5. 6.

Your Turn: Calculate the period of the following graphs: f(t) = sin(3t) 8. f(t) = sin(–4t) 9. 10. f(t) = 4sin(2t) 11. 12.

Midline A line that horizontal bisects (cuts in half) a trigonometric graph Represented by a dashed line Not part of the graph 2 2π –2

Finding the Midline Determine the height of the graph. Divide the height by 2. (This is the midline’s distance from the maximum or minimum of the graph.) Sketch the midline on the graph. f(t) = sin(t) + 2

Examples f(t) = 4sin(t) – 1 f(t) = –3sin(t) + 1

Your Turn: For problems 13 – 16 in the guided notes, solve for the midline and sketch it on the graph.

13. 14. 15. 16.

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

Calculating Amplitude Examples 17. f(t) = 6sin(4t) 18. f(t) = –5sin(6t) 19. 20.

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.

Sketching Sine Graphs – Single Smooth Line!!!