Problem 25 - Algebra 2

Question

State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. $$ y=\csc \theta-\frac{3}{4} $$

Step-by-Step Solution

Answer

Vertical shift: Down by \(\frac{3}{4}\); midline: \(y = -\frac{3}{4}\); amplitude: N/A; period: \(2\pi\).

Step 1: Identify the vertical shift

The given function is \(y = \csc \theta - \frac{3}{4}\). The term \(-\frac{3}{4}\) indicates that the function has been shifted vertically downward by \(\frac{3}{4}\) units.

Step 2: Determine the equation of the midline

For the function \(y = \csc \theta - \frac{3}{4}\), the midline is generally obtained by considering any constant terms outside the basic function. Thus, the equation of the midline is \(y = -\frac{3}{4}\), which is a horizontal line.

Step 3: Understand the amplitude

In a cosecant function, amplitude is not typically defined, since \(\csc \theta\) is not a bounded function like sine or cosine. Therefore, we say amplitude is not applicable for \(\csc \theta\).

Step 4: Calculate the period

The basic period of \(\csc \theta\) (or \(\sin \theta\)) is \(2\pi\). Since there are no horizontal stretch or compressions applied (the coefficient of \(\theta\) is 1), the period remains \(2\pi\).

Step 5: Graph the function

To graph \(y = \csc \theta - \frac{3}{4}\), recognize that this is a vertical shift of the basic \(\csc \theta\) function. Its vertical asymptotes and key points will be shifted downward by \(\frac{3}{4}\) on the y-axis. The midline at \(y = -\frac{3}{4}\) serves as a reference. Plot this along with the general shape of \(\csc \theta\) with its transformations.

Key Concepts

Vertical Shift

In trigonometric functions, a vertical shift occurs when the entire graph of the function moves up or down along the y-axis. This can be recognized by a constant being added or subtracted from the function. In the given function, \(y = \csc \theta - \frac{3}{4}\), the term \(-\frac{3}{4}\) signifies a downward shift of \(\frac{3}{4}\) units.

• A positive constant causes an upward shift.
• A negative constant, as in this case, results in a downward shift.

Understanding vertical shifts help in predicting the overall change in the position of the function on the graph, making it easier to sketch accurately.

Midline of Trigonometric Function

The midline of a trigonometric function is a horizontal line that represents the average value of the function's maximum and minimum values. It's a key reference line in understanding vertical shifts and oscillations. In the equation \(y = \csc \theta - \frac{3}{4}\), the midline is given by the equation of the constant term, which is \(y = -\frac{3}{4}\).

• The midline effectively splits the graph into two equal portions, with the oscillations of the function occurring around it.
• In functions with no vertical shift, the midline often rests on the x-axis (\(y = 0\)).

By identifying the midline, students can effectively determine the vertical position of the function in relation to the x-axis.

Period of Trigonometric Function

The period of a trigonometric function refers to the interval it takes for the function to complete one full cycle before repeating. For the function \(y = \csc \theta\), which derives from \(y = \sin \theta\), the period is the same as that of the sine function, namely \(2\pi\).

• A standard trigonometric function without any modifications will have its standard period.
• If the function's angle is modified by a coefficient \(k\), such as \(\csc(k\theta)\), the period changes to \(\frac{2\pi}{k}\).

For \(y = \csc \theta - \frac{3}{4}\), since there isn’t any such coefficient affecting \(\theta\), the period stays at \(2\pi\). Knowing the period helps in plotting the oscillations correctly on the graph.

Amplitude of Trigonometric Function

Amplitude is a measure of how much a trigonometric function varies from its midline to its peak. It describes the height of the oscillations and is crucial for understanding how "tall" the waves in the function appear. However, for a cosecant function like \(\csc \theta\), the amplitude is considered not applicable.

• Amplitude doesn't apply since the function's values are not confined between a maximum and minimum; \(\csc \theta\) extends infinitely as it approaches its asymptotes.
• In contrast, functions like sine and cosine have amplitudes, as they are bounded between -1 and 1 without vertical shifts.

Recognizing when amplitude is applicable and when it is not is crucial in understanding different types of trigonometric functions.