Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.5 Part 4 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.

Vertical Shifts The vertical movement of a graph up or down from its original position is called a “vertical shift”. A vertical shift of a circular function occurs when a term is added to, or subtracted from, the basic sine or cosine function. When a positive number is added to the basic sine or cosine function the vertical shift is up. When a positive number is subtracted from the basic sine or cosine function the vertical shift is down. In both these cases the effect is to move the “x-axis” from y = 0 to y = the amount of the vertical shift.

Vertical Shifts The vertical shift is caused by the constant d in the equations y = d + a sin(bx – c) and y = d + a cos(bx – c). The shift is d units up for d > 0 and d units down for d < 0. In other words, the graph oscillates about the horizontal line y = d instead of about the x-axis. The maximum value of y is d + |a|. The minimum value of y is d – |a|.

Example: Vertical Shift in the Sine Function Compared with y = sin x, the graph y = 1 + sin x will be moved 1 unit up. The amplitude will still be “1” and the period will still be 2π. There will be no phase shift

Example: Sketch the graph of y = 1 + sin x The primary period of this function will occur when: 0 ≤ x ≤ 2π Divide this interval into quarter points: Assign pattern of sine values (0, 1, 0, -1, 0), with “1” added to each, the five key points are: Connect the points with a smooth curve:

Example: Sketch the graph of y = 1 + sin x Points to be connected with smooth curve:

Example: Graph one period of the function y = 2cosx + 1 (same as y = 1 + 2cos x). Identify the amplitude, period, phase shift, and vertical shift. y = acos(bx – c) + d or y = d + acos(bx – c) y = 2cosx + 1 amplitude: |a| = |2| = 2 period: 2π/b = 2π/1 = 2π phase shift: none vertical shift: d = 1 unit up

Example: y = 2cosx + 1 Find the values of x for the five key points.

Example: y = 2cosx + 1 Find the values of y for the five key points.

Example: y = 2cosx + 1 3. (cont) Find the values of y for the five key points.

Example: y = 2cosx + 1 Connect the five key points with a smooth curve and graph one complete cycle of the given function.

Your Turn: Graph 2  2 sin 3x Sketch the graph of 2  2 sin 3x.

Your Turn: Sketch the graph y = 1 + 2 sin (4x + ).