Problem 31 - Algebra 2

Question

State the vertical shift, amplitude, period, and phase shift for each function. Then graph the function. $$ y=6 \cot \left[\frac{2}{3}\left(\theta-90^{\circ}\right)\right]+0.75 $$

Step-by-Step Solution

Answer

Vertical shift: 0.75; period: 270°; phase shift: 135° right; no amplitude.

Step 1: Identify Vertical Shift

The vertical shift is represented by the constant added to the function. In the given function, it is the value outside of the cotangent expression. Therefore, the vertical shift is 0.75.

Step 2: Determine Amplitude

Cotangent functions do not have an amplitude since they are not bounded. Thus, the concept of amplitude does not apply to this function.

Step 3: Calculate Period

The period of the cotangent function is determined by the coefficient in front of the variable inside the function. The standard period of a cotangent function is \(180^{\circ}\), and it is adjusted by dividing it by the coefficient of \(\theta\). In the function \( y = 6 \cot \left[ \frac{2}{3}(\theta - 90^{\circ}) \right] \), the coefficient is \(\frac{2}{3}\), so the period is \(\frac{180^{\circ}}{\frac{2}{3}} = 270^{\circ}\).

Step 4: Find Phase Shift

The phase shift is determined by the horizontal translation inside the cotangent function. The expression is \(\frac{2}{3}(\theta - 90^{\circ})\), which suggests a shift of \(90^{\circ}\) to the right, since it is \( (\theta - 90^{\circ}) \). We need to adjust this by the coefficient \(\frac{2}{3}\): \(90^{\circ} \div \frac{2}{3} = 135^{\circ}\). Therefore, the phase shift is \(135^{\circ}\) to the right.

Step 5: Graph the Function

To graph the function, begin with the basic shape of the cotangent function, which starts from positive values and decreases to negative values within its period. Shift the graph horizontally to the right by \(135^{\circ}\) and vertically upwards by 0.75 units. The period is \(270^{\circ}\), so ensure the repeating nature of the function aligns accordingly.

Key Concepts

Vertical Shift

In trigonometry, a vertical shift occurs when a constant is added or subtracted from a function, causing it to move up or down on the coordinate plane. In the function we're examining, \( y=6 \cot \left[ \frac{2}{3}(\theta-90^{\circ})\right]+0.75\), we notice the \(+0.75\) at the end. This indicates the function is shifted vertically by 0.75 units upwards.

Vertical shifts do not change the

• shape
• period
• or amplitude (if applicable) of the graph.

This is different from transformations that affect the overall characteristics of the function, such as its periodicity or symmetry. When graphing, you can imagine moving the entire curve up by 0.75 units.

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is one of the six basic trigonometric functions. It is the reciprocal of the tangent function. When exploring \( \cot(x) \), it's important to understand its properties:

• It is undefined at multiples of \(180^{\circ}\) (or \(\pi\) radians),
• Has vertical asymptotes at these points,
• And its graph resembles a repeating wave.

Unlike sine or cosine functions, cotangent does not have a maximum or minimum, which is why we do not discuss amplitude in relation to it.
In the function given, \( y=6 \cot \left[ \frac{2}{3}(\theta-90^{\circ})\right]\), the \(6\) is a scale factor affecting the steepness. However, it doesn't alter the periodic nature or the absence of an amplitude in \( \cot(x) \). Understanding these intrinsic properties is crucial when plotting cotangent or predicting its behavior.

Period and Phase Shift

The characteristics of trigonometric functions, such as period and phase shift, make them unique. The period of a function is the length over which it completes one full cycle and repeats. For standard cotangent functions, the period is \(180^{\circ} \) or \(\pi \) radians.

In the given function, \( y=6 \cot \left[ \frac{2}{3}(\theta-90^{\circ})\right]\), we modify the period by using the coefficient \(\frac{2}{3} \). Calculating it lets us know that the period becomes \(\frac{180^{\circ}}{\frac{2}{3}} = 270^{\circ}\). This means the function takes a longer distance along the x-axis to complete its full cycle.

The phase shift, on the other hand, involves a horizontal movement. In the expression \(\frac{2}{3}(\theta-90^{\circ})\), \(\theta-90^{\circ}\) indicates the initial shift of \(90^{\circ}\) to the right. Adjusting this by the coefficient \(\frac{2}{3}\), we get a true shift of \(135^{\circ}\) to the right. This modifies the starting point of the cycle on the graph. When graphing, you need to consider both these transformations for accuracy.