 # Day 7 - Graphing by Addition of Ordinates

## Introduction

Objective: To graph functions that are the sums or differences of two sine functions, two cosine functions, or a sine and a cosine function.

"The function y = 2x + 4 is actually the sum of two functions, y = 2x and y = 4. For every value in the domain, there is a corresponding value in the range so that the point (x, 2x + 4) is on the graph of y = 2x + 4. The sums of trigonometric functions can also be graphed."

Resource: Hayden, Jerome D. and Hall, Bettye C. (1993). Trigonometry. Prentice Hall: Englewood Cliffs, New Jersey.

## Student Activity

I. Graphing by addition of ordinates on Graphing Calculator.

1. Graph y = sinx over the interval [-2p, 2p]. Then, on the same coordinate plane (but in a different color), graph y = cosx over the same interval. Finally, on the same coordinate plane (but in a different color), graph y = sinx + cosx over the same interval. For each value in the domain, how do the values of the range of y = sinx + cosx compare with the values of the ranges of y = sinx and y = cosx?

2. Graph y = 2cosx over the interval [-2p, 2p]. Then, on the same coordinate plane (but in a different color), graph y = cos2x over the same interval. Finally, on the same coordinate plane (but in a different color), graph y = 2cosx + cos2x over the same interval. For each value in the domain, how do the values of the range of y = 2cosx + cos2x compare with the values of the ranges of y = 2cosx and y = cos2x?

II. Graphing by addition of ordinates by hand.

1. Open an Excel file. Label the first column "x-coordinates," the second column "y = 2sinx," the third column "y = -sin0.5x," and the fourth column "y = 2sinx - sin0.5x." Set each column to round to four decimal places.

2. Set up the first column to range from -2p to 2p by intervals of p/4.

3. Fill in the second and third columns by utilizing the equations y = 2sinx and y =0.5sinx, respectively.

4. Fill in the fourth column by simply adding the second and third columns.

5. Graph y = 2sinx + sin0.5x by using the x-coordinates from column A and the y-coordinates from column D.

## Teacher Key for Student Activity

I.

1. y = sinx is in red, y = cosx is in blue, and y = sinx + cosx is in purple Answer: At each x-coordinate, the y-coordinates of y = sinx + cosx is the sum of the y-coordinates of y = sinx and y = cosx.

2. y = 2cosx is in red, y = cos2x is in blue, and y = 2cosx + cos2x is in purple Answer: At each x-coordinate, the y-coordinates of y = sinx + cosx is the sum of the y-coordinates of y = sinx and y = cosx.

II.

2. In cell A2, enter "=-2*pi()". In cell A3, enter "=A2+pi()/4". Drag to fill down to row 18.

3. In cell B2, enter "=2*sin(A2)". Drag to fill down to row 18. In cell C2, enter "=-1*sin(A2/2)". Drag to fill down to row 18.

4. In cell D2, enter "=B2+C2". Drag to fill down to row 18.

Here is a sample table:

 x-coordinates y = 2sinx y = -sin0.5x y = 2sinx - sin0.5x -6.2832 0 0 0 -5.4978 1.4142 0.3827 1.7969 -4.7124 2 0.7071 2.7071 -3.9270 1.4142 0.9239 2.3381 -3.1416 0 1 1 -2.3562 -1.4142 0.9239 -0.4903 -1.5708 -2 0.7071 -1.2929 -0.7854 -1.4142 0.3827 -1.0315 0 0 0 0 0.7854 1.4142 -0.3827 1.0315 1.5708 2 -0.7071 1.2929 2.3562 1.4142 -0.9239 0.4903 3.1416 0 -1 -1 3.9270 -1.4142 -0.9239 -2.3381 4.7124 -2 -0.7071 -2.7071 5.4978 -1.4142 -0.3827 -1.7969 6.2832 0 0 0

5. Here is a sample graph: ## Student Practice

Graph each of the following functions (by hand!) over the interval [-2p, 2p].

1. y = sinx + 2cosx

2. y = 3sinx + sin2x

3. y = 2sinx - cosx

4. y = cos2x + sin2x

## Teacher Key for Student Practice

1. 2. 3. 4. 