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To graph the function [tex]\( y = 0.5 \sec \left( x + \frac{\pi}{3} \right) - 2 \)[/tex], we need to understand the components that make up this equation. Let's break it down step-by-step:
1. Understanding the Basic Secant Function:
The secant function, [tex]\( \sec(x) \)[/tex], is the reciprocal of the cosine function:
[tex]\[ \sec(x) = \frac{1}{\cos(x)} \][/tex]
The secant function has vertical asymptotes where the cosine function is zero, which occurs at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], for any integer [tex]\( k \)[/tex].
2. Phase Shift:
The function [tex]\( \sec \left( x + \frac{\pi}{3} \right) \)[/tex] represents a horizontal shift of [tex]\( \sec(x) \)[/tex] to the left by [tex]\( \frac{\pi}{3} \)[/tex].
3. Amplitude Change:
The coefficient [tex]\( 0.5 \)[/tex] scales the secant function vertically. The maximum and minimum values of [tex]\( \sec(x) \)[/tex] are scaled by [tex]\( 0.5 \)[/tex], making the range of [tex]\( 0.5 \sec(x) \)[/tex] to be [tex]\( (-\infty, -0.5] \cup [0.5, \infty) \)[/tex].
4. Vertical Shift:
Subtracting 2 from the function shifts the entire graph downward by 2 units. This changes the range of [tex]\( 0.5 \sec \left( x + \frac{\pi}{3} \right) - 2 \)[/tex] to be [tex]\( (-\infty, -2.5] \cup [-1.5, \infty) \)[/tex].
5. Plotting Key Points:
- Start by plotting the vertical asymptotes. For [tex]\( \sec \left( x + \frac{\pi}{3} \right) \)[/tex], these occur where [tex]\( \cos \left( x + \frac{\pi}{3} \right) = 0 \)[/tex], which is at:
[tex]\[ x + \frac{\pi}{3} = \frac{\pi}{2} + k\pi \implies x = \frac{\pi}{6} + k\pi \quad (\text{for any integer } k) \][/tex]
So the asymptotes occur at [tex]\( x = \frac{\pi}{6} + \pi k \)[/tex].
- The typical points of [tex]\( \sec(x) \)[/tex] where it takes on maximum and minimum values also need to be adjusted for the phase shift.
6. Sketching the Graph:
- Notice that between the vertical asymptotes [tex]\( \frac{\pi}{6} + k\pi \)[/tex], the [tex]\( \sec \left( x + \frac{\pi}{3} \right) \)[/tex] could take maximum and minimum values.
- Apply the vertical stretch by 0.5.
- Shift the graph down 2 units.
### Graph Key Characteristics:
1. Vertical Asymptotes: At [tex]\( x = \frac{\pi}{6} + k\pi \)[/tex].
2. Maximum Points: For [tex]\( x = n\pi - \frac{\pi}{3} \)[/tex] where [tex]\( n \)[/tex] is even, the maximum will be [tex]\( 0.5 - 2 = -1.5 \)[/tex].
3. Minimum Points: For [tex]\( x = n\pi - \frac{\pi}{3} \)[/tex] where [tex]\( n \)[/tex] is odd, the minimum will be [tex]\( -0.5 - 2 = -2.5 \)[/tex].
### Conclusion
The graph of [tex]\( y = 0.5 \sec \left( x + \frac{\pi}{3} \right) - 2 \)[/tex] should exhibit vertical asymptotes at [tex]\( x = \frac{\pi}{6} + n\pi \)[/tex], maximum values of -1.5, and minimum values of -2.5. It systematically exhibits periodicity and algebraic manipulation assurance for exact characteristics mapping.
1. Understanding the Basic Secant Function:
The secant function, [tex]\( \sec(x) \)[/tex], is the reciprocal of the cosine function:
[tex]\[ \sec(x) = \frac{1}{\cos(x)} \][/tex]
The secant function has vertical asymptotes where the cosine function is zero, which occurs at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], for any integer [tex]\( k \)[/tex].
2. Phase Shift:
The function [tex]\( \sec \left( x + \frac{\pi}{3} \right) \)[/tex] represents a horizontal shift of [tex]\( \sec(x) \)[/tex] to the left by [tex]\( \frac{\pi}{3} \)[/tex].
3. Amplitude Change:
The coefficient [tex]\( 0.5 \)[/tex] scales the secant function vertically. The maximum and minimum values of [tex]\( \sec(x) \)[/tex] are scaled by [tex]\( 0.5 \)[/tex], making the range of [tex]\( 0.5 \sec(x) \)[/tex] to be [tex]\( (-\infty, -0.5] \cup [0.5, \infty) \)[/tex].
4. Vertical Shift:
Subtracting 2 from the function shifts the entire graph downward by 2 units. This changes the range of [tex]\( 0.5 \sec \left( x + \frac{\pi}{3} \right) - 2 \)[/tex] to be [tex]\( (-\infty, -2.5] \cup [-1.5, \infty) \)[/tex].
5. Plotting Key Points:
- Start by plotting the vertical asymptotes. For [tex]\( \sec \left( x + \frac{\pi}{3} \right) \)[/tex], these occur where [tex]\( \cos \left( x + \frac{\pi}{3} \right) = 0 \)[/tex], which is at:
[tex]\[ x + \frac{\pi}{3} = \frac{\pi}{2} + k\pi \implies x = \frac{\pi}{6} + k\pi \quad (\text{for any integer } k) \][/tex]
So the asymptotes occur at [tex]\( x = \frac{\pi}{6} + \pi k \)[/tex].
- The typical points of [tex]\( \sec(x) \)[/tex] where it takes on maximum and minimum values also need to be adjusted for the phase shift.
6. Sketching the Graph:
- Notice that between the vertical asymptotes [tex]\( \frac{\pi}{6} + k\pi \)[/tex], the [tex]\( \sec \left( x + \frac{\pi}{3} \right) \)[/tex] could take maximum and minimum values.
- Apply the vertical stretch by 0.5.
- Shift the graph down 2 units.
### Graph Key Characteristics:
1. Vertical Asymptotes: At [tex]\( x = \frac{\pi}{6} + k\pi \)[/tex].
2. Maximum Points: For [tex]\( x = n\pi - \frac{\pi}{3} \)[/tex] where [tex]\( n \)[/tex] is even, the maximum will be [tex]\( 0.5 - 2 = -1.5 \)[/tex].
3. Minimum Points: For [tex]\( x = n\pi - \frac{\pi}{3} \)[/tex] where [tex]\( n \)[/tex] is odd, the minimum will be [tex]\( -0.5 - 2 = -2.5 \)[/tex].
### Conclusion
The graph of [tex]\( y = 0.5 \sec \left( x + \frac{\pi}{3} \right) - 2 \)[/tex] should exhibit vertical asymptotes at [tex]\( x = \frac{\pi}{6} + n\pi \)[/tex], maximum values of -1.5, and minimum values of -2.5. It systematically exhibits periodicity and algebraic manipulation assurance for exact characteristics mapping.
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