Problem 30 - Algebra 2

Question

State the vertical shift, amplitude, period, and phase shift for each function. Then graph the function. $$ y=3 \csc \left[\frac{1}{2}\left(\theta+60^{\circ}\right)\right]-3.5 $$

Step-by-Step Solution

Answer

Vertical shift: down 3.5, amplitude influence: 3, period: 720°, phase shift: 30° left.

Step 1: Identify the Vertical Shift

The equation is given as \( y = 3 \csc\left[ \frac{1}{2}(\theta + 60^\circ) \right] - 3.5 \). The constant term at the end, \(-3.5\), indicates the vertical shift. This means the graph is shifted downward by 3.5 units.

Step 2: Determine the Amplitude

The amplitude is not directly applicable to the cosecant function like it is for sine or cosine. However, the coefficient 3 in front of the \( \csc \) function affects the stretching of the graph vertically. Although traditionally we say that cosecant doesn't have an amplitude, you can think of \(|3|\) as influencing the vertical dilation.

Step 3: Calculate the Period

The period of the \( \csc \) function is determined by the coefficient of \( \theta \). The general formula for the period of \( \csc(b \theta) \) is \( \frac{360^\circ}{b} \). Here, \( b = \frac{1}{2} \), so the period is \( \frac{360^\circ}{1/2} = 720^\circ \). This means the function repeats every 720 degrees.

Step 4: Find the Phase Shift

The phase shift is determined by the expression within the cosecant function, \( \frac{1}{2}(\theta + 60^\circ) \). Rewriting it as \( \frac{1}{2}\theta + 30^\circ \) shows the horizontal shift: 30 degrees to the left. This is calculated by setting \( \frac{1}{2}(\theta + 60^\circ) = 0 \), solving for \( \theta \).

Step 5: Graph the Function

To graph \( y = 3 \csc\left[ \frac{1}{2}(\theta + 60^\circ) \right] - 3.5 \), start by sketching the reciprocal sine function: \( y = 3 \sin\left[ \frac{1}{2}(\theta + 60^\circ) \right] \). Draw its asymptotes where the sine function has zeros. Shift the entire graph downward by 3.5 units and introduce vertical stretching due to the coefficient 3.

Key Concepts

Vertical Shift

The concept of a vertical shift involves moving a graph up or down along the y-axis. It is represented by the constant added or subtracted from the function. In our equation \( y = 3 \csc\left[ \frac{1}{2}(\theta + 60^\circ) \right] - 3.5 \), the vertical shift is determined by the \(-3.5\). This means we will shift the entire graph downward by 3.5 units.

• A positive constant moves the graph up.
• A negative constant moves the graph down.

Understanding vertical shifts is crucial as they alter the baseline from which the graph's peaks and troughs are measured. In this case, everything on the graph is synchronically shifted downward, including the middle line and asymptotes.

Period of Cosecant Function

The period of a cosecant function refers to the interval it takes for the function to complete one full cycle and start repeating its pattern. For the cosecant function \( \csc(b\theta) \), the period is found using the formula \( \frac{360^\circ}{b} \). In this function, \( b \) is \( \frac{1}{2} \), resulting in a period of \( \frac{360^\circ}{1/2} = 720^\circ \).

• Period tells how wide the repeating cycle of the function is.
• This determines the distance between repeated portions of the function's graph.

The setting of the period helps you predict where the pattern will reoccur, making it easier to plot and understand the graph of the function.

Phase Shift

The phase shift involves moving the trigonometric graph horizontally along the x-axis. In the function \( y = 3 \csc\left[ \frac{1}{2}(\theta + 60^\circ) \right] \), a phase shift is observed through the expression \( \frac{1}{2}(\theta + 60^\circ) \). By rewriting it as \( \frac{1}{2}\theta + 30^\circ \), we reveal a phase shift of 30 degrees to the left.

• Phase shift of \(+\) indicates a leftward move.
• Phase shift of \(-\) indicates a rightward move.

Calculating the phase shift helps in aligning the graph to its correct horizontal positioning, ensuring that features like peaks match the intended points along the x-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions involves a few important steps to ensure accuracy. For the equation \( y = 3 \csc\left[ \frac{1}{2}(\theta + 60^\circ) \right] - 3.5 \), you begin by considering its reciprocal sine function.

• Sketch the graph of \( y = 3 \sin\left[ \frac{1}{2}(\theta + 60^\circ) \right] \).
• Identify regions where sine is zero, marking asymptotes on the graph for the \( \csc \) function.
• Apply the identified vertical and phase shifts.
• Introduce vertical stretching due to the coefficient 3 in front of the sine or resulting cosecant function.

Graphing combines all these elements to visualize the function correctly, accounting for shifts and transformations, enabling a deeper understanding of how trigonometric functions behave.