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

### 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 . In radians, the coordinate for the tangent function would be

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

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 and . If we were to extend the graph out in either direction, there would continue to be vertical asymptotes at the odd multiples of . Therefore, the domain is all real numbers, , where 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 where and are the same as they are for the other trigonometric functions. For simplicity, we will not address phase shifts in this concept.

Let's graph from and state the domain and range.

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

Notice that the vertical asymptotes did not change. The **period** of this function is still . 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 .

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

Now, let's graph from , state the domain and range, and find all zeros within this domain.

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

The domain is all real numbers, where is any integer. The range is all real numbers. To find the zeros, set .

Finally, let's graph from and state the domain and range.

This function has a period of . The domain is all real numbers, except , where is any integer. The range is all real numbers.

### Examples

#### Example 1

Earlier, you were asked to find the period and zeros of the function .

The period is .

The zeros are where is zero.

#### Example 2

Find the period of the function .

The period is .

#### Example 3

Find the zeros of the function from Example 2, from .

The zeros are where is zero.

#### Example 4

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

The general equation is . We know that . Let’s use the period to solve for the frequency, or .

The equation is .

### Review

Graph the following tangent functions over . Determine the period, domain, and range.

- Find the zeros of the function from #1.
- Find the zeros of the function from #3.
- Find the zeros of the function from #5.

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

- Amplitude: 3 Period:
- Amplitude: Period:
- Amplitude: -2.5 Period: 8
**Challenge**Graph over . Determine the domain and period.

### Answers for Review Problems

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