Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.5 Graphs of Sine and Cosine Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Understand the graph of y = sin x. Understand the graph of y = cos x. Graph variations of y = sin x. Graph variations of y = cos x. Use vertical shifts of sine and cosine curves. Model periodic behavior.

Modeling Periodic Behavior To write the equation of a sinusoidal function from a graph you have to; 1st - determine the type of function it is, Sine or Cosine. 2nd - calculate the a, b, c, and d in the equations y = a sin (bx – c) + d or y = a cos (bx – c) + d.

Modeling Periodic Behavior To determine the function, Sine or Cosine; Find the midline or axis of the curve and the start point of the curve. The start point of the curve will be the y-axis if there is no phase shift and to the right or left of the y-axis if there is a phase shift. If the curve begins on the midline at the start point, the function is a sine curve. If the curve begins above or below the midline at the start point, the function is a cosine curve.

Modeling Periodic Behavior To calculate a, b, c, and d; a (amplitude) = distance from min. to max. / 2 (also determine sign, + if starts above midline and – if starts below midline). b = 2π/period (find the period from the graph) c = b(phase shift) find the phase shift from the graph d = (max. + min.)/2 Substitute into the equation y = a sin (bx – c) + d or y = a cos (bx – c) + d.

Example #1: Write the equation of the graph. To start, we need to find the midline. We can find this by finding the vertical distance between a high point and a low point. One high point is (5, 2). One low point is (0, -2). This distance between the two is the absolute value of the difference between the y-coordinates or |2-(-2)| = 4. The midline is half way between these two point or 2 away for either. Two units below a high point is at 0. This gives us a midline of y = 0.

Example #1: (continued) Now we look at the intersection of the midline and the y-axis. If the graph passes through this intersection, then the equation is most easily written as a sine equation. If the graph has a high point or a low point at the y-axis, then the equation is most easily written as a cosine equation. Our graph has a low point at the y –axis, thus we will use a basic cosine equation.

Example #1: (continued) Now we just need to fill in numbers for a, b, c, and d. We already know that d is 0 since the midline is y = 0. We also know that c is 0 since we are going to attempt to write the equation of graph as a cosine function without a phase shift.

Example #1: (continued) Now, find a, a will need to be negative since the basic cosine graph has a high point on the y-axis. Our graph has been reflected over the x-axis and thus has a low point on the y-axis. The |a| is the distance between the midline and a high point. The midline is y = 0 and a high point is (5, 2). Thus |a| = |2-0| = 2. Thus a is -2.

Example #1: (continued) Now find b. The period of the function is (2π)/b. We can find the period by counting the number of units between two low points on this graph. Here the period is 10 units. Take the period of 10 and plug it into the equation to solve for b. b = 2π/period = 2π/10 = π/2

Example #1: (continued) Thus our equation is

Find an equation of the graph to the right. Example #2: Find an equation of the graph to the right. Solution The graph looks like a sine wave. So its equation is of the form y = a sin (bx – c) + d. Determine the constants a, b, c, and d from the graph. The graph appears in the sine form, so a = 3.

Example #2: (continued) By inspection, the period of the graph is 6. We know that

Example #2: (continued) © Example #2: (continued) The graph starts at (1, 2). So the phase shift is 1. Therefore, The average of the maximum value and the minimum value gives the vertical shift d for the sinusoidal graph.

Example #2: (continued) © Example #2: (continued) Substitute the values of a, b, c, and d in y = a sin (bx – c) + d to get the equation.

Example 6 Your Turn: DETERMINING AN EQUATION FOR A GRAPH Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph.