Analyzing the Graph that y = tan x and also Its Variations

We will start with the graph of the tangent function, plotting points as we did because that the sine and also cosine functions. Remind that


The period of the tangent duty is π because the graph repeats itself on intervals of where k is a constant. If us graph the tangent duty on −fracpi2\ to fracpi2\, we deserve to see the actions of the graph top top one finish cycle. If us look at any larger interval, us will view that the qualities of the graph repeat.

You are watching: Using complete sentences, explain the key features of the graph of the tangent function.

We have the right to determine whether tangent is one odd or even function by making use of the definition of tangent.

eginarray � an(−x)=fracsin(−x)cos(−x) hfill& extDefinition the tangent. \ =frac−sin xcos x hfill& extSine is an odd function, cosine is even. \ =−fracsin xcos x &hfill extThe quotient of one odd and also an even function is odd. hfill \ =− an x hfill& extDefinition that tangent. endarray\

Therefore, tangent is an odd function. We can further analyze the graphical actions of the tangent role by spring at values for some of the one-of-a-kind angles, as noted in the table below.

x −fracpi2\−fracpi3\−fracpi4\−fracpi6\0fracpi6\fracpi4\fracpi3\fracpi2\
tan (x) undefined−sqrt3\–1−fracsqrt33\0fracsqrt33\1sqrt3\undefined

These clues will aid us draw our graph, but we must determine just how the graph behaves where it is undefined. If us look more closely in ~ values when fracpi3 x 1.31.51.551.56 tan x 3.614.148.192.6

As x philosophies fracpi2\, the outputs of the role get larger and larger. Since y= an x\ is one odd function, we watch the corresponding table of an adverse values in the table below.

x −1.3−1.5−1.55−1.56
tan x −3.6−14.1−48.1−92.6

We have the right to see that, together x philosophies −fracpi2\, the outputs get smaller and smaller. Remember that there space some worths of x because that which cos x = 0. For example, cosleft(fracpi2 ight)=0\ and cosleft(frac3pi2 ight)=0\. At these values, the tangent function is undefined, so the graph that y= an x has actually discontinuities at x=fracpi2\ and also frac3pi2\. At these values, the graph of the tangent has vertical asymptotes. Number 1 represents the graph of y= an x\. The tangent is hopeful from 0 to fracpi2\ and from π come frac3pi2\, matching to quadrants I and III that the unit circle.


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Figure 2


Because A=0.5 and also B=fracpi2\, we can discover the stretching/compressing factor and also period. The duration is fracpifracpi2=2\, therefore the asymptotes are at x=pm 1. At a quarter duration from the origin, us have

eginarrayf(0.5)=0.5 an(frac0.5pi2) hfill& \ =0.5 an(fracpi4) hfill& \ =0.5 endarray\

This way the curve must pass v the points(0.5,0.5),(0,0),and(−0.5,−0.5).The just inflection allude is in ~ the origin. Figure shows the graph the one period of the function.


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Figure 4


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As us did because that the tangent function, we will again refer to the consistent |A| as the extending factor, not the amplitude.


Similar to the secant, the cosecant is identified by the reciprocal identity csc x=1sin x. An alert that the function is undefined when the sine is 0, causing a vertical asymptote in the graph at 0, π, etc. Since the sine is never an ext than 1 in pure value, the cosecant, being the reciprocal, will never ever be less than 1 in pure value.

We deserve to graph y=csc x through observing the graph the the sine function because these two features are reciprocals of one another. See figure 10. The graph that sine is shown as a dashed orange tide so we deserve to see the relationship. Whereby the graph the the sine function decreases, the graph the the cosecant function increases. Whereby the graph the the sine role increases, the graph the the cosecant role decreases.

The cosecant graph has vertical asymptotes at each worth of x whereby the sine graph crosses the x-axis; we present these in the graph below with dashed vertical lines.

Note that, because sine is one odd function, the cosecant function is also an weird function. That is, csc(−x)=−csc x.

The graph of cosecant, which is presented in number 10, is comparable to the graph the secant.

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Figure 19


57. A video camera is concentrated on a rocket top top a launching pad 2 mile from the camera. The angle of key from the ground to the rocket after ~ x secs is fracpi120x.

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a. Create a duty expressing the altitude h(x), in miles, of the rocket above the ground after ~ x seconds. Overlook the curvature the the Earth.b. Graph h(x) ~ above the term (0,60).c. Evaluate and interpret the worths h(0) and also h(30).d. What wake up to the values of h(x) as x approaches 60 seconds? interpret the definition of this in regards to the problem.