Graphing Secant and Cosecant - Trigonometry
Card 0 of 8
This is the graph of what function of x?
This is the graph of what function of x?
is shown in red, and 
is shown in blue.
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Give the equation of the following graph.
Give the equation of the following graph.
Looking at our graph, we can tell that the period is 
. Using the formula
where 
is the coefficient of 
and 
is the period, we can calculate
This eliminates one answer choice. We then retrun to our graph and see that the amplitude is 3. Remembering that the amplitude is the number in front of the function, we can eliminate two more choices.
We then examine our graph and realize it contains the point 
. Plugging 0 into our two remaining choices, we can determine which one gives us 4 for a result.
Looking at our graph, we can tell that the period is
This eliminates one answer choice. We then retrun to our graph and see that the amplitude is 3. Remembering that the amplitude is the number in front of the function, we can eliminate two more choices.
We then examine our graph and realize it contains the point
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Which of the following is the graph of 
?
Which of the following is the graph of 
In order to graph 
, recall that 
. First consider the graph 
.
Now anywhere this graph crosses the x-axis a vertical asymptote will form for the 
graph because the denominator of 
will be equal to zero and the function will be undefined. At each maximum and minimum of 
, the graph of 
will invert at that point.
And then we are left with the graph of 
.
In order to graph 
Now anywhere this graph crosses the x-axis a vertical asymptote will form for the 
And then we are left with the graph of 
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Considering the general form of the cosecant transformation function 
, what does each letter, (A, B, C, and D) correspond to?
Considering the general form of the cosecant transformation function 
Since cosecant is a reciprocal of sine, it uses the same general formula of the sine function with the letters corresponding to the same transformations. Note that while A does correspond to amplitude, the cosecant function extends infinitely upwards and downwards so there is no amplitude for the graphs.
Since cosecant is a reciprocal of sine, it uses the same general formula of the sine function with the letters corresponding to the same transformations. Note that while A does correspond to amplitude, the cosecant function extends infinitely upwards and downwards so there is no amplitude for the graphs.
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Which of the following is the graph of 
?
Which of the following is the graph of 
Knowing that the graph of 
is
we can use the general form of the cosecant transformation equation, 
, and apply these transformations. We can ignore 
because in this case 
. In this case 
and so our period is:
Period = 
Period = 
Period = 
is the normal period for cosecant graphs and so we do not have to worry about lengthening or shortening the period. 
and so we need to apply a phase shift of 
. This will cause our graph to shift left a total of 
units.
Lastly, we must apply the transformation for 
, so we will have an upward vertical shift of 1 unit.
The application of these transformations leaves us with our graph of 
.
Knowing that the graph of 
we can use the general form of the cosecant transformation equation, 
Period =
Period =
Period =
Lastly, we must apply the transformation for 
The application of these transformations leaves us with our graph of 
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Which of the following is the graph for 
?
Which of the following is the graph for 
In order to understand the graph of secant, recall that 
. First consider the graph of 
.
Anywhere this graph crosses the x-axis a vertical asymptote will form for the 
graph because the denominator of 
will be equal to zero and the function will be undefined. At each maximum and minimum of 
, the graph of 
will invert at that point.
And then we are left with the graph 
.
In order to understand the graph of secant, recall that 
Anywhere this graph crosses the x-axis a vertical asymptote will form for the 
And then we are left with the graph 
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True or False: Amplitude must be considered when graphing the transformation of a secant graph.
True or False: Amplitude must be considered when graphing the transformation of a secant graph.
When looking at the graph of 
, it extends infinitely upwards and downwards from each local maximum and minimum. This will be true for all transformed secant graphs as well. Due to this, there is no amplitude for secant graphs. However, secant is the reciprocal of cosine graphs which do rely on amplitude for transformations. For this reason amplitude must be considered as a vertical shift.
When looking at the graph of 
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Which of the following is the graph of 
?
Which of the following is the graph of 
Knowing that the general form of the graph 
is:
We can use the general form of the cosecant transformation equation, 
, and apply these transformations. 
because secant graphs extend infinitely upwards and downwards and does not have an amplitude, we must think of the secant graph being a reciprocal of the cosine graph. So we will consider 
for cosine.
We will shift our secant graph to invert at the maximums and minimums of the cosine graph.
Next, we will factor 
in order to get our equation into the form 
.
And so 
. We can now solve for our period,
Period = 
Period = 
Period = 
This shortens our original period of 
to 
.
Now we must consider 
. This will give us a phase shift of 
units to the left. Since our period has also been shortened this does not change the graph visually. 
in this case so we do not need to consider a vertical shift.
And we are left with the graph of 
.
Knowing that the general form of the graph 
We can use the general form of the cosecant transformation equation, 
We will shift our secant graph to invert at the maximums and minimums of the cosine graph.
Next, we will factor 
And so 
Period =
Period =
Period =
This shortens our original period of 
Now we must consider 
And we are left with the graph of 
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