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2.7.2: Tangent Graphs

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Adjust the length of the curve, or the distance before the y values repeat, from 2π.

Your mission, should you choose to accept it, as Agent Trigonometry is to find the period and the zeros of the function y=12tan4x.

Graph of a Tangent Function

The graph of the tangent function is very different from the sine and cosine functions. First, recall that the tangent ratio is tanθ=oppositeadjacent. In radians, the coordinate for the tangent function would be (θ,tanθ)

xθ0π6π4π3π22π33π43π6πytanθ03313und. 31330

After π, the y-values repeat, making the tangent function periodic with a period of π.

f-d_c03ec795c9f6253ca74dd700e4841fc869a57f5afbe8870bca27198c+IMAGE_TINY+IMAGE_TINY.png
Figure 2.7.2.1

The red portion of the graph represents the coordinates in the table above. Repeating this portion, we get the entire tangent graph. Notice that there are vertical asymptotes at x=3π2, π2, π2 and 3π2. If we were to extend the graph out in either direction, there would continue to be vertical asymptotes at the odd multiples of \pi 2. Therefore, the domain is all real numbers, xnπ±π2, where n is an integer. The range would be all real numbers. Just like with sine and cosine functions, you can change the amplitude, phase shift, and vertical shift.

The standard form of the equation is y=atanb(xh)+k where a, b, h, and k are the same as they are for the other trigonometric functions. For simplicity, we will not address phase shifts (k) in this concept.

Let's graph y=3tanx+1 from [2π,2π] and state the domain and range.

First, the amplitude is 3, which means each y-value will be tripled. Then, we will shift the function up one unit.

f-d_07407024505dc0d8c6c8fbb1544c9d1d46d3768d018b51c88d2185f1+IMAGE_TINY+IMAGE_TINY.png
Figure 2.7.2.2

Notice that the vertical asymptotes did not change. The period of this function is still π. Therefore, if we were to change the period of a tangent function, we would use a different formula than what we used for sine and cosine. To change the period of a tangent function, use the formula πb.

The domain will be all real numbers, except where the asymptotes occur. Therefore, the domain of this function will be xR, xnπ±π2. The range is all real numbers.

Now, let's graph y=tan2π from [0,2π], state the domain and range, and find all zeros within this domain.

The period of this tangent function will be π2 and the curves will be reflected over the x-axis.

f-d_6ded33339f78bfa3428c3fa88b70bbe6b52956251c9c767d0f86fe94+IMAGE_TINY+IMAGE_TINY.png
Figure 2.7.2.3

The domain is all real numbers, xπ4,3π4,5π4,7π4,π4±π2n where n is any integer. The range is all real numbers. To find the zeros, set y=0.

0=tan2x0=tan2x2x=tan10=0,π,2π,3π,4πx=0,π2,π,3π2,2π

Finally, let's graph y=14tan14x from [0,4π] and state the domain and range.

This function has a period of π14=4π. The domain is all real numbers, except 2π,6π,10π,2π±4πn, where n is any integer. The range is all real numbers.

f-d_93802e42950eec259d55d26c1f669378464e24e78f6d9339ff5ab472+IMAGE_TINY+IMAGE_TINY.png
Figure 2.7.2.4
Example 2.7.2.1

Earlier, you were asked to find the period and zeros of the function y=12tan4x.

Solution

The period is π4.

The zeros are where y is zero.

0=12tan4x0=tan4x4x=tan10=0,π,2π,3πx=14(0,π,2π,3π)x=0,π4,π2,34π

Example 2.7.2.2

Find the period of the function y=4tan32x.

Solution

The period is π32=π23=2π3.

Example 2.7.2.3

Find the zeros of the function from Example 2, from [0,2π].

Solution

The zeros are where y is zero.

0=4tan32x0=tan32x32x=tan10=0,π,2π,3πx=23(0,π,2π,3π)x=0,2π3,4π3,2π

Example 2.7.2.4

Find the equation of the tangent function with an amplitude of 8 and a period of 6π.

Solution

The general equation is y=atanbx. We know that a=8. Let’s use the period to solve for the frequency, or b.

πb=6πb=π6π=16

The equation is y=8tan16x.

Review

Graph the following tangent functions over [0,4π]. Determine the period, domain, and range.

  1. y=2tanx
  2. y=13tanx
  3. y=tan3x
  4. y=4tan2x
  5. y=12tan4x
  6. y=tan12x
  7. y=4+tanx
  8. y=3+tan3x
  9. y=1+23tan12x
  10. Find the zeros of the function from #1.
  11. Find the zeros of the function from #3.
  12. Find the zeros of the function from #5.

Write the equation of the tangent function, in the form y=atanbx, with the given amplitude and period.

  1. Amplitude: 3 Period: 3π2
  2. Amplitude: 14 Period: 2π
  3. Amplitude: -2.5 Period: 8
  4. Challenge Graph y=2tan13(x+π4)1 over [0,6π]. Determine the domain and period.

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.5.

Additional Resource

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