Lauren is playing with a yo-yo and keeping track of its height over time. She has plotted some key points on the graph at right. The $y$-values are in inches and the $x$-values are in seconds. Complete the following problems based on the graph.

1. How long does it take to complete one cycle?

   Hint (a):

   

2. How long is the string?

   Hint (b):

   

3. Find an equation that will model the height of the yo-yo over time.

   Hint (c):

   Find:
   1. Amplitude
   2. Vertical Shift
   3. Horizontal Shift
   4. $b$ (frequency)
   
   Note: part (a) is the period and $pb = 2π$
   Test your answer in your graphing calculator.

4. Find the first two times when the yo-yo was exactly $30$ inches above the ground.

   Hint (d):

   Find where the function crosses $y = 30$.
   Note: The first two times may have occurred before the curve in the given sketch.