7.2More on Graphs of Sine and Cosine
OBJECTIVE1: Sketching Graphs of the Form and
The Graphs of
The amplitude is (since ) and the range is .
The period is .
The phase shift is C.
The x-coordinates of the quarter points are .
OBJECTIVE 2: Sketching Graphs of the Form and
Steps for Sketching Functions of the Form and
Step 1:Rewrite the function as or . If, then use the even and odd properties of the sine and cosine function to write the function in an equivalent form such that .
We now use this new form to determine the amplitude,period, and phase shift.
Step 2: The amplitude is . The range is .
Step 3: The period is.
Step 4: The phase shift is .
Step 5: The x-coordinate of the first quarter point is . The x-coordinate of the last quarter point is .
An interval for one complete cycle is . Subdivide this interval into 4 equal subintervals of length by starting with and adding to the x-coordinate of each successive quarter point.
Step 6: Multiply the y-coordinates of the quarter points of or by Ato determine
the y-coordinates of the corresponding quarter points for and
.
Step 7: Connect the quarter points to obtain one complete cycle.
OBJECTIVE 3: Sketching Graphs of the Form and
Steps for Sketching Functions of the Form and
Step 1:Rewrite the function as or . If, then use the even and odd properties of the sine and cosine function to write the function in an equivalent form such that .
We now use this new form to determine the amplitude, period, and phase shift.
Step 2: The amplitude is.The range is .
Step 3: The period is.
Step 4: The phase shift is .
Step 5: The x-coordinate of the first quarter point is . The x-coordinate of the last quarter point is .
An interval for one completecycle is . Subdivide this interval into 4 equal subintervals of length by starting with and adding to the x-coordinate of each successive quarter point.
Step 6: Multiply the y-coordinates of the quarter points of or by Athen add D to determine
the y-coordinates of the corresponding quarter points for and
.
Step 7: Connect the quarter points to obtain one complete cycle.
OBJECTIVE 4Determine the Equation of a Function of the Form or Given Its Graph
Suppose that you were asked to determine the equation of a function that describes the graph
seen in the figure below where the function is of the form or .
Furthermore, suppose that you were asked to choose a function that describes the graph seen in the figure where the function is chosen from the five functions listed below.
Which of the five functions above would you choose to accurately describe the graph displayed in the figure? The answer is all five functions describe the graph seen in the figure! You might want to take a few minutes and indeed verify that all of these functions describe the same graph.
Therefore, given the graph of a sine curve or cosine curve, it is impossible to determine a unique function that describes the graph unless certain assumptions are made. In this objective, when given the graph of a function of the form or , we will first state whether the given graph is a sine curve or a cosine curve and we will assume that . We will also always label the five quarter points of one cycle of the given graph to correspond with the five quarter points of the graph of or
over the interval . If these assumptions are made, then we can determine a unique function that describes the given graph.
We can follow the six steps outlined below to determine the function that describes a given sine or cosine curve.
Steps for Determining an Equation of a Function of the Form
or Given the Graph
Step 1.Subtract the x- coordinate of the first quarter point from the x-coordinate of the fifth quarter point to determine the period.
Step 2.Use the equation to determine the value of .
Step 3.The x-coordinate of the first quarter point represents the phase shift, . Use this information to determine the value of C.
Step 4.The amplitude is where is the range of the given graph.
Step 5.If the given graph is a sine curve, then if the graph is increasing from the first quarter point to the second quarter point. Otherwise, .
If the given graph is a cosine curve, then if the graph is decreasing from the first quarter point to the second quarter point. Otherwise, .
Step 6.Choose the first quarter point that lies on the given graph and the corresponding first quarter point of the graph of or . Note that the first quarter point of is and the first quarter point of is .
The relationship between , the y-coordinate of the first quarter point of the given graph, and the y-coordinate of the first quarter point of or is given by the following:
If the graph is a sine curve, then.
If the graph is a cosine curve, then .
Use this information to determine the value of D.