Write the equation of the graph below AS A COSINE FUNCTION??

enter image source here

1 Answer
Sean
Apr 10, 2018

#y=2cos8(theta-pi/8)+3#

Explanation:

.

Let's take a look at the graph of a cosine function:

enter image source here

At #theta=0 and 2pi#, #y=1#

At #theta=pi#, #y=-1#

The amplitude of the function is #1#.

The period of the function is #2pi#

The graph is symmetrical with respect to the #y#-axis.

The general form of a cosine function is:

#color(red)(y=acosb(theta+-c)+-d)#

Where #a=# amplitude, #b=# angle coefficient, #c=# horizontal shift (phase shift), and #d=# vertical shift.

Comparing the problem graph with this graph, we notice that if we start with this function as the base function we will have to implement a horizontal and a vertical shift.

If we draw a horizontal line at a point that makes the graph symmetrical with respect to the line we will see that the part of the graph above the line has a height of #2#. Therefore,

#a=2#

We can see that the problem graph has one full period between #0# and #pi/4#. We calculate this period by dividing #2pi# by the coefficient of the angle in the problem function. Therefore,

#(2pi)/b=pi/4#

#b=8#

The first maximum of the problem graph happens at #pi/8#. This means the graph has been shifted to the right (positive direction) by #pi/8#. Therefore,

#c=pi/8#

If we move the graph down vertically by #3# units it becomes symmetrical with respect to the #y#-axis. This means the graph has been shifted up by #3# units. Therefore,

#d=3#

The equation of our function is:

#y=2cos8(theta-pi/8)+3#

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