1. 14.2 Translations and Reflections of Trigonometric Graphs
Algebra 2

2. Graphing Sine and Cosine Functions
To obtain the graph of y = a sin b(x – h) + k or y = a cos b(x – h) + k transform (move) the graphs of y = a sin bx or y = a cos bx as follows.
Vertical Shift- Shift the graph k units vertically
Horizontal Shift – Shift the graph h units horizontally
Reflection- If a<0, reflect the graph in the line y=k after the vertical and horizontal shifts

3. Examples: Write an equation for the graph describing the following.
The graph of shifted up 4 units.
The graph of shifted left 4 units.
The graph of reflected over the x-axis.

4. More Examples

5. Examples:
Graph
Graph y=-cos4x

6. Example:
Graph

7. More Examples:

8. Example:
Give the amplitude, period, and five key points of the graph of

9. Example:
You are riding a Ferris wheel. You height h (in feet) above the ground at any time t (in seconds) can be modeled by the equation:
The Ferris wheel turns for 160 sec before it stops to le the first passenger off.

10. Example (continue) Sketch the graph of your height with respect to t.
What are your minimum and maximum heights?

11. Example: On another Ferris wheel, your height s given by the equation
How many cycles will this Ferris wheel make in 150sec?
What are your maximum and minimum heights?
What is you height after 150sec?

12. Graphing Tangent Functions
To obtain the graph of y = a tan b(x – h) + k, transform y = a tan bx as follows
Shift the graph k units vertically and h units horizontally.
Then, if a<0, reflect the graph in the line y=k.

13. Examples:
Graph
Give the asymptotes, the halfway points, and center points of the graph of y=2 – 3tan 2x

14. More Examples:

15. Example:
You are standing 90 ft from where a balloon was launched. The balloon travels straight up to a maximum height of 120 ft. What is the angle of elevation of the balloon when it is 50 ft from the maximum height?

16. Example:
A balloon is launched 150 ft away from you. It can reach a maximum height o 200 ft. What is the angle of elevation of this balloon when it is 80 ft from the maximum height?