In this case, we add C and D to the general form of the tangent function. The graph of a transformed tangent function is different from the basic tangent function tan x in several ways:.
Step 1. Step 3. Step 4. Step 5—7. The graph is shown in Figure 4. Step 2. Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions.
Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function. We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period. This means that every 4 seconds, the beam of light sweeps the wall.
The distance from the spot across from the police car grows larger as the police car approaches. See Figure 8. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value. See Figure 9. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases.
When the cosine function is zero, the secant is undefined. The secant graph has vertical asymptotes at each value of x where the cosine graph crosses the x -axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant. Note that, because cosine is an even function, secant is also an even function. As we did for the tangent function, we will again refer to the constant A as the stretching factor, not the amplitude.
Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value. See Figure The graph of sine is shown as a dashed orange wave so we can see the relationship.
Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases. The cosecant graph has vertical asymptotes at each value of x where the sine graph crosses the x -axis; we show these in the graph below with dashed vertical lines. Note that, since sine is an odd function, the cosecant function is also an odd function. That is, we locate the vertical asymptotes and also evaluate the functions for a few points specifically the local extrema.
If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left.
Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions. The equations become the following. Step 5. Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function. Steps 6—7. We can use two reference points, the local minimum at 0, 2. Figure 11 shows the graph. Step 6. There is a local minimum at 1. Figure 12 shows the graph. The excluded points of the domain follow the vertical asymptotes.
Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function. Figure 13 shows the graph. What are the domain and range of this function? The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots.
Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.
We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following. Plot two reference points. Step 7. Step For the following exercises, rewrite each expression such that the argument x is positive. For the following exercises, sketch two periods of the graph for each of the following functions.
Identify the stretching factor, period, and asymptotes. For the following exercises, find and graph two periods of the periodic function with the given stretching factor, A , period, and phase shift. What is the function shown in the graph? Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let x , measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and x is measured negative to the left and positive to the right.
A laser rangefinder is locked on a comet approaching Earth. A video camera is focused on a rocket on a launching pad 2 miles from the camera. Thinking back to when you learned about graphing rational functions , a zero in the denominator means you'll have a vertical asymptote. Let's put dots for the zeroes and dashed vertical lines for the asymptotes:.
Now we can use what we know about sine, cosine, and asymptotes to fill in the rest of the tangent's graph: We know that the graph will never touch or cross the vertical asymptotes; we know that, between a zero and an asymptote, the graph will either be below the axis and slide down the asymptote to negative infinity or else be above the axis and skinny up the asymptote to positive infinity. Since sine and cosine are periodic, then tangent has to be, as well.
A quick check of the signs tells us how to fill in the rest of the graph:. The Tangent Graph. The concept of "amplitude" doesn't really apply. Then draw in the curve. You can plot a few more points if you like, but you don't generally gain much from doing so. If you prefer memorizing graphs, then memorize the above. But I always had trouble keeping straight anything much past sine and cosine, so I used the reasoning demonstrated above to figure out the tangent and the other trig graphs.
As long as you know your sines and cosines very well, you'll be able to figure out everything else.
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