# The Sine Function

## Warm up

Press the play button or move the slider.

What is the x-value of the red point?

What is the y-value of the red point?

Tracing the red points results in the graph of .
This sine function relates an angle measure in radians to the y-value of a point on the unit circle.

What type of function is the sine function?

## Transforming Sine Functions

Move the sliders.

How does changing the slider for *a* affect the graph of this sine function?

What happens to the graph when *a* is negative?

How does changing the slider for *b* affect the graph of this sine function?

## Key Concept (Take Note!)

For the periodic function , where , , and is an angle in radians:

- The amplitude of the function is .
- The function has cycles between 0 and .
- The period of the function is .

Which function would have an amplitude of 4?

Which function would have a period of pi?

## Sketching a Sine Function

- Draw an x-axis from 0 to the period of the function. Divide the period into 4 pieces.
- Draw a y-axis from the amplitude to the opposite of the amplitude.
- Use a five-point summary to sketch the graph of a sine function:

*zero-max-zero-min-zero*if

*a*is positive.

* zero-min-zero-max-zero*if

*a*is negative.

Sketch the graph of .

Write an equation for the sine function shown above.

## Exercise!

pg. 848 - 849
#15 - 30 all, 47,
48