Let's break down the given equation \( y = 0.5 \sec \left( x + \frac{\pi}{3} \right) - 2 \) step by step to understand its key characteristics. This will help us identify the correct graph.
### Step 1: Understanding the Basic Secant Function
The basic secant function, \( \sec(x) \), is defined as the reciprocal of the cosine function, \( \sec(x) = \frac{1}{\cos(x)} \). The secant function has vertical asymptotes where the cosine function is zero (i.e., \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer).
### Step 2: Amplitude and Vertical Stretch
The factor 0.5 in \( 0.5 \sec(x) \) scales the secant function vertically by 0.5. This means the secant function’s range, which is normally \( (-\infty, -1] \cup [1, \infty) \), will now become \( (-\infty, -0.5] \cup [0.5, \infty) \).
Amplitude: 0.5
### Step 3: Phase Shift
The term \( x + \frac{\pi}{3} \) indicates a horizontal shift of the graph of \( \sec(x) \). Specifically, the horizontal shift for \( \sec \left( x + \frac{\pi}{3} \right) \) is \( -\frac{\pi}{3} \) (left by \(\frac{\pi}{3}\)).
Phase Shift: \( -\frac{\pi}{3} \approx -1.0471975511965976 \)
### Step 4: Vertical Shift
The final term \( -2 \) is a vertical translation, shifting the entire function downward by 2 units.
Vertical Shift: -2
### Step 5: Period of the Secant Function
The normal period of the secant function \( \sec(x) \) is \( 2\pi \). Since there is no horizontal stretching/compression factor (other than shifting), the period remains \( 2\pi \).
Period: \( 2\pi \approx 6.283185307179586 \)
### Summary of Key Characteristics:
- Amplitude: 0.5
- Phase Shift: \( -\frac{\pi}{3} \approx -1.0471975511965976 \)
- Vertical Shift: -2
- Period: \( 2\pi \approx 6.283185307179586 \)
### Identifying the Correct Graph:
The graph of \( y = 0.5 \sec \left( x + \frac{\pi}{3} \right) - 2 \) will:
1. Have vertical asymptotes shifted left by \( \frac{\pi}{3} \).
2. Be scaled vertically by a factor of 0.5.
3. Shift downward by 2 units.
4. Repeat with a period of \( 2\pi \).
Using these characteristics, you should be able to identify the correct graph among the options given. Look for a secant graph that has these specific transformations applied.
