Writing Equations of Tangent Functions

Equations of Tangent Functions

The tangent function is a type of periodic function that has both an amplitude and a period. Remember that amplitude is the distance from the center to the highest or lowest point of the graph. The period is the distance between two respective points on a wave.

The tangent line equation is: $f(x) = tan(x)$

The tangent function is a discontinuous graph, meaning that you have to pick up your pencil to draw the graph on a coordinate plane. The period of a tangent function is $\pi$.

Here is a graph of a tangent function:

The domain of this function is complicated because of the discontinuity of the graph. The domain includes all real numbers of $x$, except for the values with vertical asymptotes. On a standard, parent tangent function, the vertical asymptotes are at: $x = -π/2$, $x=π/2$, $x=-3π/2$, $x=3π/2$, and so on. The function will never touch these asymptotes. The range of this function is all real numbers.

To translate the function horizontally, we can consider the following formula: $y=tan(x+h)$, where $h$ represents the horizontal translation. This may look familiar to you - it kind of looks like point slope form!

Even if you are given partial information about the tangent function, you can use the above information to help in finding the equation.

Equations of the Tangent Function

The equation of the tangent function is set up very similarly to the sine and cosine functions.

The standard form of the tangent function is:

$$y=a \text{ tan }b(x-h)+k$$

$|a|$ is the amplitude of the function. $b$ helps us find the period, or horizontal compression. $h$ tells us the horizontal translation, while $k$ tells us the vertical translation.

Writing the Equation for the Tangent Graph

by WOWmath

In this video by WOWmath you learn how to write a tangent function equation, given a tangent graph.

Summary

We are given the following tangent graph:

The first thing we are going to look for on our tangent graph is the midline. We can determine the midline by looking for the point of inflection, which is where our graph seems to switch direction. In this graph, it looks like the midline is on the $x$-axis, $y=0$.

Now that we have the midline, we should think about where the period is. The period starts and ends at the same place, and then it repeats. For this graph, the period is $\frac{\pi}{2}$. Since the tangent equation doesn't directly use the period, we need to calculate the $b$ value.

$$b=\pi \div \frac{\pi}{2} = \frac{\pi}{1} \cdot \frac{2}{\pi} = 2$$

Next, we will look at the midpoint between the origin and halfway to the first vertical asymptote. It looks like our vertical stretch is $\frac12$, and the graph is flipped.

We now have all of the information we need to write the equation!

$$y=\frac12 \text{ tan }(2x)$$

He goes through one more example - try it on your own, then watch the video to see how you did!