Problem 45 - Algebra and Trigonometry

Question

In Exercises 45–52, graph two periods of each function. $$ y=2 \tan \left(x-\frac{\pi}{6}\right)+1 $$

Step-by-Step Solution

Answer

The graph of \(y = 2 \tan (x - \frac{\pi}{6}) + 1\) is a transformation of the standard tangent function that is stretched vertically by a factor of 2, shifted horizontally to the right by \(\frac{\pi}{6}\) units, and shifted upwards by 1 unit. It has a period of \(\pi\), undefined amplitude, a vertical shift of 1, and a horizontal shift of \(\frac{\pi}{6}\). The graph repeats every \(\pi\) units.

Step 1: Identifying the Equation

The given equation \(y = 2 \tan (x - \frac{\pi}{6}) + 1\) is a transformed version of the standard tangent function. It has a vertical stretch factor of 2, horizontal shift of \(\frac{\pi}{6}\) units to the right and a vertical shift of 1 unit up.

Step 2: Finding the Period

The period of the tangent function is \(\pi\). The horizontal shift does not affect the period, so the period of the given tangent function is also \(\pi\). The function will repeat after an interval of \(\pi\).

Step 3: Finding the Amplitude

The amplitude of the normal tangent function is not defined because the range of the function extends from minus infinity to plus infinity. The range of the given function will also do the same, so the amplitude doesn't have a defined value.

Step 4: Defining the Vertical Shift

There is a vertical shift of 1 unit upwards. This indicates that all the points on the graph of the tangent function are lifted up by 1 unit.

Step 5: Defining the Horizontal Shift

There is a horizontal shift of \(\frac{\pi}{6}\) units to the right meaning that each point on the graph of the ordinary tangent function is shifted \(\frac{\pi}{6}\) units to the right.

Step 6: Graphing the Function

After determining the period, vertical shift, and horizontal shift, graph the function using these properties. Start by graphing the original \(\tan(x)\) function, then adjust the graph as per the transformations. Multiply the y-values by 2 (vertical stretch), shift the graph right by \(\frac{\pi}{6}\) units (horizontal shift), and then shift it upward by 1 unit (vertical shift). The graph will then contain two complete periods of the function.

Key Concepts

Tangent Function Period

Understanding the period of a tangent function is crucial when graphing it. In its simplest form, the tangent function, written as \(y = \tan(x)\), has a period of \(\pi\). This means the values of the function repeat every \(\pi\) radians. For the exercise at hand, the period of the function \(y = 2 \tan(x - \frac{\pi}{6}) + 1\) remains \(\pi\) despite the transformations because period adjustments would come from coefficients in front of \(x\), but not from horizontal shifts.

In other words, within every interval of length \(\pi\), the tangent function goes through its entire range - from negative to positive infinity - before looping back to its starting value. While graphing, you can think of the period as the 'width' of one cycle of the function, showing you where to repeat the pattern as you move along the x-axis.

Horizontal Shift

Horizontal shifts occur in functions when there is an addition or subtraction within the argument of the function. This kind of transformation is also known as a phase shift. In the exercise, the term \(x - \frac{\pi}{6}\) is responsible for a horizontal shift. Particularly, this indicates a shift to the right by \(\frac{\pi}{6}\) units.

When graphing \(y = 2 \tan (x - \frac{\pi}{6}) + 1\), every point on the curve of the standard tangent function \( \tan(x)\) is moved \(\frac{\pi}{6}\) units to the right. Visually, this shift can be understood by imagining sliding the graph of \( \tan(x)\) along the x-axis, without altering its shape. This results in the start and end points of each period being \(\frac{\pi}{6}\) sizes away from their original position, thus affecting the locations of all the characteristic points of the tangent function, such as its zeros and asymptotes.

Vertical Shift

Vertical shifts in functions happen when a constant is added or subtracted from the function itself. This moves the graph up or down. In the given function \(y = 2 \tan (x - \frac{\pi}{6}) + 1\), the '+1' at the end of the equation indicates a vertical shift of 1 unit upwards.

This important transformation takes every single point on the original tangent curve and lifts it up by 1 unit. This includes both the x-axis intercepts (which turn into intercepts at \(y=1\)) and the asymptotes (which also move up by 1 unit). During graphing, after applying the horizontal shift, the final step is to adjust for this vertical lift, ensuring all points are accurately positioned 1 unit higher than they would be on the standard \( \tan(x)\) curve.