At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
