>> We're given the trig function Y equals 3 times tangent
of the quantity 2X plus pi divided by 2.
We want to find the period and the horizontal shift and then graph the function so for review
if we have the tangent or cotangent function in this form here.
Remember A is not the amplitude.
Tangent and cotangent have no amplitude though the value
of A really they're vertically stretched
or vertically compressed the graph of tangent and cotangent.
Next the period of the function is equal to pi divided by B. Notice how this quantity here is
in factored form and we have pi here not 2 pi because remember pi is a period
of the basic tangent and cotangent function.
Following their function this is 2 pi divided by B and then C the constant
on the end vertically translates or vertically shifts the graph.
So if C is positive the graph is shifted up C units and if the C is negative,
if we have minus a constant here then the graph is shifted down the opposite value of C units
and then finally D, this value here horizontally shifts or horizontally translates the function.
If D is greater than zero meaning we have X minus D then the function is shifted right D
units and if D is negative, so we'd have X plus D then the function is shifted the opposite
value of D units to the left.
So going back to our example the first thing you need to recognize is that we have
to factor this quantity here so that we have 2 times X plus some quantity.
We want the coefficient of X to be one.
So starting with the given function we would have Y equals 3 times tangent
of if we factor out 2 from these two terms.
We factor out 2 from 2 X we'd have X. If we factor out 2 from pi divided by 2 we can think
of dividing by 2 and multiplying by 1/2 that would be 5 divided by 4.
Notice how if we distribute the 2 we have 2 pi divided by 4 which is still pi over 2.
Notice in this form it can use 2 to find the period and we can use this pi
over 4 to find the horizontal shift.
So again, the period would be equal to pi divided by B
which in this case it would be pi divided by 2 and then the horizontal shift,
notice here we have X plus pi divided by 4
which means the graph has actually shifted left pi over 4 radians.
Now we are going to go ahead and grab this function but notice how the value
of A is positive 3 that's not the amplitude
but this will vertically stretch the graph of our function.
To graph the transformation of this tangent function though we're going to rely
on key components of the basic tangent function graphed here.
Notice if we focus on one piece of a tangent function, let's say the piece
between negative pi over 2 radians and positive pi over 2 radians notice
over the period the graph starts and ends as a vertical asymptote.
In the middle of the period the function value is zero this points here.
If we divide this into four equal parts here and here notice the function value 1/4 of the period
to the right of the center, here the function value is one and notice if you go 1/4
of the period to the left of the center here the function value is negative one.
We'll use these vertical asymptotes and these three key points to graph this transformation.
Let's begin by focusing on this point here in the middle of the period
where the function value is zero.
If our graph is shifted pi however 4 units
to the left we'll take this point here and shift it left pi 4 radians.
So instead of being at the origin if you label this negative pi over 4 radians
that point would now be here and from here since the period is pi over 2 radians
if we divide this in half that would be pi over 4 radians
which means the vertical asymptotes would be pi over 4 radians to the right and pi 4 radians
to the left at this point in the center which means we'll have a vertical asymptotes here pi
over 4 radians to the left and here pi over 4 radians to the right of the center.
So this would be zero radians and this would be negative 2 pi divided
by 4 radians or negative pi over 2 radians.
Now let's go ahead and divide this interval into fourths.
So we'll divide this here and we'll divide this here.
Figure if you go 1/4 of the period to the right
of the center this function value is positive one but because of this coefficient of 3 here
when this tangent function value is positive one it will be 3 times one or positive 3.
So figure 1/4 right of the center the function value would be positive 3 here not positive one.
So again, this is 1, 2, 3, negative 1, negative 2 and negative 3.
So now figure out 1/4 of the period to the left of the center,
instead of having a function value of negative 1, it'll be 3 times negative 1 or negative 3.
Again, the co-efficient of 3 is vertically stretching this graph and now our graph passes
through these three points and approaches asymptote so it would look like this.
Now that we have one piece of this graph we can just copy
and paste this to the right and to the left.
So let's go ahead and divide this up into its periods.
Here and here would be the vertical asymptotes on the right.
On the left we have one here.
The function values in the middle of each of these intervals would be zero
so we have a point here, here, and here.
Divide this into fourths.
If we move 1/4 of the period to the right
of center the function value would be positive 3 so here, here and here.
If we move 1/4 to the left of the center the function value would be negative 3.
Now we can sketch more of the graph so here's a graph of the given function.
I hope you found this helpful. ------------------------------dfaf2be9e889--