Problem 11 - Algebra 2

Question

State the vertical shift, amplitude, period, and phase shift for each function. Then graph the function. $$ y=\frac{1}{2} \sec \left[4\left(\theta-\frac{\pi}{4}\right)\right]+1 $$

Step-by-Step Solution

Answer

Vertical shift: 1 unit up; No amplitude; Period: \(\frac{\pi}{2}\); Phase shift: \(\frac{\pi}{4}\) right.

Step 1: Identifying the Vertical Shift

In the equation \(y=\frac{1}{2} \sec \left[4\left(\theta-\frac{\pi}{4}\right)\right]+1\), the term outside the trigonometric function adds a constant to the whole function, which is \(+1\). This means there is a vertical shift of 1 unit upwards.

Step 2: Determining the Amplitude

The amplitude is typically associated with sine and cosine functions, indicating half the distance between the maximum and minimum values. Since secant does not have a maximum or minimum value due to its nature following cosine's reciprocal, it does not have an amplitude in the traditional sense. The transformation affects scaling instead.

Step 3: Calculating the Period

For a secant function of the form \(y=a \sec(b(\theta-c))+d\), the period is calculated using the formula \(\frac{2\pi}{b}\). Here, \(b=4\), so the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).

Step 4: Identifying the Phase Shift

The phase shift is determined by the horizontal translation within the trigonometric function, given by \(c\) in the form \(b(\theta-c)\). Here, \(c = \frac{\pi}{4}\), resulting in a rightward phase shift of \(\frac{\pi}{4}\) units.

Key Concepts

Vertical Shift

In trigonometric functions, a **vertical shift** moves the graph of a function up or down. This happens when a constant is added outside the function. For the function \(y=\frac{1}{2} \sec \left[4\left(\theta-\frac{\pi}{4}\right)\right]+1\), the '+1' at the end indicates the entire graph is shifted upwards by 1 unit.

• This does not affect the shape of the graph; the entire function simply starts 1 unit higher on the y-axis than it normally would.
• Vertical shifts can be positive, moving the graph upwards, or negative, moving it downwards.

Understanding vertical shifts is crucial when sketching graphs as it helps you position the function correctly on the coordinate plane.

Secant Function

The **secant function** is a trigonometric function that is the reciprocal of the cosine function. It is denoted as \(\sec(\theta)\).

• The secant function is undefined where cosine is zero because division by zero is undefined. This creates vertical asymptotes at these points.
• Instead of crossing the x-axis like sine or cosine, the secant function creates u-shaped curves opening upwards or downwards.
• This function tends to repeat its pattern every time cosine completes a full cycle.

In the given function, notice that we begin with \(\frac{1}{2}\sec(...)\), which scales the graph of the secant function. Even though amplitude isn't classically defined for secant, this scaling factor affects how far the arms of the U shapes stretch vertically.

Period

The **period** of a trigonometric function is the interval required for the function to complete one full cycle. For secant functions expressed in the form \(y=a \sec(b(\theta-c))+d\), the period is calculated as \(\frac{2\pi}{b}\).

• In the exercise's equation, we find \(b=4\).
• Thus, the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).
• This means the pattern of the secant function will repeat every \(\frac{\pi}{2}\) units along the x-axis.

Understanding the period helps you predict where the asymptotes and curves will appear on the graph, making it easier to draw or analyze.

Phase Shift

The **phase shift** refers to the horizontal movement of a trigonometric graph. It's determined by shifts in the angle within the trigonometric function. In our secant function \(b(\theta-c)\), this shift is impacted by the value \(c\).

• If \(c\) is positive, the graph shifts to the right.
• If \(c\) is negative, the graph shifts to the left.
• In this function, \(c = \frac{\pi}{4}\), which results in a rightward phase shift.

Phase shifts are vital for understanding how the initial position of the graph is altered in relation to the y-axis, ensuring an accurate representation on the coordinate plane.