Lesson 3

In Try This 2, you may have noticed that the parameters a, b, c, and d in y = a cos b(x − c) + d describe the same characteristics as they did for y = a sin b(x − c) + d.

So far, you have worked with sine and cosine functions in degrees. It is also possible to use an x-value in radians. All the parameters behave the same, but now c is in radians. The relationship between b and the period also changes slightly: 360° = 2π, so use  instead of when working in radians.

The following example shows how an equation in radians can be interpreted.

Example

Harry examined the function and asked the following questions.

1. What is the equation of the midline?
2. What is the period of the graph?
3. Describe any horizontal translation of the graph from y = cos x.
4. What are the maximum and minimum values?

Solution

1. The midline has the equation y = d. This means the equation of the midline is y = 2.
2. The equation is written in radians because the c-value has no unit.
   
   
    
   360° = 2π rad
   
   So, the equation  can be used to determine the period of the function.
   
   
    
   
3. The c-value determines any horizontal translation from y = cos x. A −4 must have been substituted for c into y = a sin b(x − c) + d. This means the graph moved left 4 units.
4. The amplitude is equal to a and so is . The maximum occurs one amplitude above the midline, so the maximum is . The minimum occurs one amplitude below the midline, so the minimum is .
   
   The graphing calculator can graph  in order to check the solutions.
   
   
   

Self-Check 2