Analyzing the Graph and Properties of a Secant Function

Q3. Given the function $y = 4 \sec\left(\frac{1}{2}x\right)$:

• What is the stretching factor?

• What is the period? (Type pi for $\pi$, if needed.)

• Where are the asymptotes? Enter a list of values, separated by commas. Type pi for $\pi$, as needed.

Background

Topic: Graphs and Properties of Trigonometric Functions (Secant Function)

This question tests your understanding of how to analyze and graph a transformed secant function, including identifying the stretching factor, period, and vertical asymptotes.

Key Terms and Formulas

• Secant Function: $\sec(x) = \frac{1}{\cos(x)}$

• General Form: $y = a \sec(bx)$

• Stretching Factor: The value of $a$ stretches or compresses the graph vertically.

• Period of Secant: $\text{Period} = \frac{2\pi}{|b|}$

• Vertical Asymptotes: Occur where $\cos(bx) = 0$ (since $\sec(bx)$ is undefined there).

Step-by-Step Guidance

1. Identify the stretching factor by looking at the coefficient $a$ in $y = a \sec(bx)$. Here, $a = 4$.

2. Determine the period using the formula $\frac{2\pi}{|b|}$, where $b$ is the coefficient of $x$ inside the secant function. In this case, $b = \frac{1}{2}$.

3. Set up the equation for the period: $\text{Period} = \frac{2\pi}{|1/2|}$.

4. To find the vertical asymptotes, solve $\cos\left(\frac{1}{2}x\right) = 0$. Recall that $\cos(\theta) = 0$ at $\theta = \frac{\pi}{2} + n\pi$, where $n$ is any integer.

5. Set $\frac{1}{2}x = \frac{\pi}{2} + n\pi$ and solve for $x$ to find the locations of the asymptotes.

Try solving on your own before revealing the answer!