Unit 6, Lesson 15
Transforming Trigonometric Functions
Integrated Math 3
15.1
Warm-up
Match each equation with its graph. Be prepared to explain your reasoning.
A
B
C
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Match each equation with its graph. Be prepared to explain your reasoning.
A
B
C
A fan has a radius of 1 foot. A point, , starts in the position shown in the picture. The center of the fan is at and point is at the position on the circle. The fan turns in a counterclockwise direction.
Sketch a graph of the horizontal position, , in feet, of as a function of the angle of rotation, , of the fan from its starting position.
Sketch a graph of the vertical position, , in feet, of as a function of the angle of rotation, , of the fan.
The graphs of cosine and sine functions can be translated vertically or horizontally, and the size or height of their graphs can also be modified similarly to how we transformed other types of functions in an earlier unit. Let’s look at the graphs of and .
The coefficient 2 stretches the graph vertically, doubling the amplitude of the sine graph. This means that the distance from the midline to the maximum or minimum value is now 2 instead of 1. Adding 3 to the equation translates the midline up by 3 units.
What if we want to translate the graph of to the left? We can do this by adding an angle to the input, . Let’s look at the graph of . The graph of this function looks just like the graph of translated to the left by . (It also looks a lot like !)