>> We want to find the possible equation for the transformed trigonometric function.
So analyzing the graph we should recognize this as a transformation
of the secant function or cosecant function.
To help us find this transformation we'll sketch the cosine or sine function
that has the same transformation and then find the equation of the cosine or sine function
and use that equation to find the equation of our transformed secant or cosecant function.
Before we do this though we need to remember that if we were
to graph the basic secant function and cosine function
on the same coordinate plane this would be the relationship.
More specifically notice how these relative minimums of the secant function here
and here are the maximums of the cosine function and these relative maximums
of the secant function here and here are the minimums of the cosine function.
And the same thing would be true for the basic cosecant and sine functions as we see here.
So going back to our graph these 2 points will be maximums of the cosine or sine function
and these 3 points will be minimums of the sine or cosine function.
Notice how here we have Y values of positive 1, here we have Y values of negative 3.
The midline would have to be halfway between these, which would be at Y equals negative 1.
And therefore the sine or cosine function
that has the same transformation would have to look something like this.
So if we can find the equation of this transformed sine
or cosine function we can use this to find the equation
of our transformed secant or cosecant function.
So now we will focus on one period of our red graph,
so if we start at 0 radiance lets focus on this piece here.
Notice that 0 radiance we have a maximum value here in red
and therefore we'll use the cosine function to find the equation of this red graph.
Remember the 5 key points for the basic cosine function starting at 0 radiance are here
to maximum midline minimum midline and maximum.
This completes 1 period of the cosine function.
So we'll find the equation of this cosine function in the form Y equals A times cosine
of B times the quantity X minus D plus C. So we should recognize that A affects the amplitude,
B affects the period, D affects the face shift or horizontal shift
and C affects the vertical shift.
So starting with A, remember the altitude value of A it would be the amplitude
or A would be the distance from the midline to the maximum
which it'd be here notice how that's 2 units as well as it is since from the midline
to a minimum here which of course is also 2 units.
And again because that 0 radiant is going
to have a maximum function value not a minimum function value there's no reflection
across the horizontal axis and therefore A would have to be positive 2.
Now let's find B, remember 2pi divided by B is equal to the period.
So again going back over to our graph here notice how the period
of this red cosine function is 8 radiance and therefore 2pi divided by B must equal 8.
Let's write this as 8 over 1 we'll cross multiply notice how this would give us 8B
equals 2pi.
So if we divide both sides by 8, notice how we have B equals 2pi divided
by 8 or B is equal to pi over 4.
Next let's take a look at the value of D,
remember D effects the horizontal shift or face shift.
Again since that 0 radiance we have a maximum function value here, there is no face shift
and therefore D would be equal to 0.
And then finally C which represents a vertical shift
and because a midline is at Y equals negative 1.
The function is shifted down 1 unit and therefore C is equal to negative 1.
So the equation of our red cosine function they would have the same transformation
of our secant function would be Y equals 2 times cosine of again D is 0 so we'd just have BX
which it'd be pi over 4X plus C but C is negative 1 so we have minus 1.
Now again since we used the cosine function here the equation in the blue graph
that we're looking for would be the reciprocal function of cosine which is secant.
So the equation we're looking for is Y equals 2 times secant of pi over 4X minus 1.
And of course the nice way to check this would be to enter both of these equations
on a graphing calculator or some graphing software to verify the relationship
between the 2 graphs would look like this.
Both this red cosine function and this blue secant function have the same transformation.
I hope you found this helpful. ------------------------------9e4ba9d9cbea--