1. Section 5.3 Trigonometric Graphs
Chapter 5 – Trigonometric Functions: Unit Circle Approach
Section 5.3 Trigonometric Graphs
5.3 - Trigonometric Graphs

2. 5.3 - Trigonometric Graphs
Graph y = sinx
We graph a function by plotting points that satisfy that function in the coordinate system. We will graph the sine function on the period [0,2]
5.3 - Trigonometric Graphs

3. 5.3 - Trigonometric Graphs
This is called the primary cycle of the sine function.
5.3 - Trigonometric Graphs

4. 5.3 - Trigonometric Graphs
Remember that the sine function will repeat indefinitely because it is periodic so we have the graph:
5.3 - Trigonometric Graphs

5. 5.3 - Trigonometric Graphs
Properties
Domain
Range
Period
Even/Odd
x-int
Min/Max
5.3 - Trigonometric Graphs

6. 5.3 - Trigonometric Graphs
y = A sin (Bx - C) + D
We need to know what A, B, C, and D mean on the sine graph.
5.3 - Trigonometric Graphs

7. 5.3 - Trigonometric Graphs
Amplitude
Amplitude, |A|, adjusts the height of the graph and determines the range of the graph. The range of the function is [-|A|, |A|]. If |A| ≥ 1, then the graph stretches. If |A| ≤ 1, then the graph shrinks. If A < 0, then the graph reflects over the x-axis.
5.3 - Trigonometric Graphs

8. 5.3 - Trigonometric Graphs
Period
The period is (2)/B
5.3 - Trigonometric Graphs

9. 5.3 - Trigonometric Graphs
Phase Shift
The phase shift, C/B, will help us find the starting point and ending point of the primary cycle. If C/B < 0, then the graph shifts left. If C/B > 0, then the graph shifts right.
5.3 - Trigonometric Graphs

10. 5.3 - Trigonometric Graphs
Vertical Shift
Vertical shift, D, shifts the graph vertically. If D > 0, then the graph shifts up D units. If D < 0, then the graph shifts down D units. Max or min y-value is A+D or –A+D. The smaller value is the min and the large value is the max. The graph will be centered vertically on the line y = D.
5.3 - Trigonometric Graphs

11. Graphing the Sine Function
Identify the A, B, C, and D values
Identify the amplitude, period, and phase shift.
Find the values for the five key points – the three intersection points to y=D, the max, and the min.
Connect the 5 key points with a smooth curve and graph the primary cycle.
Extend the graph if needed.
5.3 - Trigonometric Graphs

12. Graphing the Negative Sine Function
When graphing the negative sine function, start off by graphing the positive sine function. Then reflect the graph over the y = D line.
NOTE: This will change your minimum and maximums.
5.3 - Trigonometric Graphs

13. 5.3 - Trigonometric Graphs
Examples
Determine the Sine equation.
5.3 - Trigonometric Graphs

14. 5.3 - Trigonometric Graphs
Examples
Determine the amplitude, period, phase shift and vertical shift of the function below. Graph the primary cycle.
5.3 - Trigonometric Graphs

15. 5.3 - Trigonometric Graphs
Graph y = cosx
We graph a function by plotting points that satisfy that function in the coordinate system. We will graph the cosine function on the period [0,2]
5.3 - Trigonometric Graphs

16. 5.3 - Trigonometric Graphs
The primary cycle is in dark blue.
5.3 - Trigonometric Graphs

17. 5.3 - Trigonometric Graphs
Properties
Domain
Range
Period
Even/Odd
x-int
Min/Max
5.3 - Trigonometric Graphs

18. 5.3 - Trigonometric Graphs
y = Acos(Bx - C) + D
Notice that we have the same values as the sine function so we will follow a similar method to graph the cosine function.
5.3 - Trigonometric Graphs

19. Graphing the Cosine Function
Identify the A, B, C, and D values
Identify the amplitude, period, and phase shift.
Find the values for the five key points –the two maxs, the min, and the two intersection points to y = D,
Connect the 5 key points with a smooth curve and graph the primary cycle.
Extend the graph if needed.
5.3 - Trigonometric Graphs

20. Graphing the Negative Cosine Function
When graphing the negative cosine function, start off by graphing the positive cosine function. Then reflect the graph over the y = D line.
NOTE: This will change your minimum and maximums.
5.3 - Trigonometric Graphs

21. 5.3 - Trigonometric Graphs
Examples
Determine the cosine equation.
5.3 - Trigonometric Graphs

22. 5.3 - Trigonometric Graphs
Examples
Determine the amplitude, period, phase shift and vertical shift of the function below. Graph the primary cycle.
5.3 - Trigonometric Graphs