Graphing Cosine Functions|Please help!

Question

1. Sketch the graph of the given function:

y = 3 cos pi/2 (x + 2) + 2

For this function, I know that my a = 3, my b = pi/2, my period is 4, my midline is 2, and that my phase shift is 2 to the left, but I am having a bit of trouble graphing it, primarily with my period. I know my graph should start at -2, but even with my period I am confused as to where the cycle should end. I used a calculator as a reference for my graph and found that the first cycle would end at 2 (I think..), why is this?

Any help would be very much appreciated!

Answer 1

Step 1: Draw the midline and the upper limit (midline plus the amplitude) and then the lower limit (midline minus the amplitude). As you mentioned, the midline is y = 2 which makes the upper limit y = 5 and lower limit y = -1:

Step 2: Draw a vertical line where the period starts (using the phase shift), so x = -2. And then draw another vertical line where the period ends, so x = 2 (-2 + 4 = 2): Also label the x-axis and y-axis.

Step 3: Within the rectangle that you have created will be where you will draw one period of the cosine function. Now, cut that rectangle in half with another vertical line right in the middle (that happens to be the y-axis. Then cut each of the two rectangles you created in half again with two more vertical lines, One would be at x = -1 and one would be at x = 1. This results in the original rectangle being divided into 4 equal spaces,

Step 4: The cosine function starts at the top so draw a dot at the upper limit at x = -2 (the starting point). Draw another dot at the "1st quarter line" (x = -1) at the midline. Draw another dot at the lower limit at the "2nd quarter line (y-axis) at the lower limit. Draw another dot at the 3rd quarter line at the midline. Draw another dot back at the upper limit at the end of the period, x = 2,

Step 5: Connect the dots with a cosine wave. If you need two periods, repeat the process to the left or to the right of your single period graph.

Answer 2

A cycle begins when x = -2 and ends when x = -2 + period = -2 + 4 = 2.

Divide the interval [-2, 2] into four equal parts and plot the maximum points, minimum point, and points that lie on the midline.