14 2 Translations and Reflections of Trigonometric Graphs

Graphing Sine and Cosine Functions � To obtain the graph of y = a

Examples: � Write an equation for the graph describing the following. ◦ The graph

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

Example: � You are riding a Ferris wheel. You height h (in feet) above

Example (continue) � Sketch to t. � What the graph of your height with

Example: � On another Ferris wheel, your height s given by the equation ◦

Graphing Tangent Functions � To obtain the graph of y = a tan b(x

Examples: � Graph � Give the asymptotes, the halfway points, and center points of

Example: � You are standing 90 ft from where a balloon was launched. The

Example: �A balloon is launched 150 ft away from you. It can reach a

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14. 2 Translations and Reflections of Trigonometric Graphs Algebra 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

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.

More Examples

Examples: � Graph y=-cos 4 x

Example: � Graph

More Examples:

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

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.

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

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

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.

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

More Examples:

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?

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?