Answer :
Certainly! Let's go through the process of understanding and graphing the function \( y = 3 \sec \left[2 \left( x - \frac{\pi}{2} \right) \right] + 2 \).
### Step-by-Step Solution:
1. Understanding the Secant Function:
The secant function, \(\sec(x)\), is the reciprocal of the cosine function, \(\cos(x)\). Thus, \(\sec(x) = \frac{1}{\cos(x)}\). The function \(\sec(x)\) has vertical asymptotes wherever \(\cos(x) = 0\), which occurs at odd multiples of \(\frac{\pi}{2}\) (i.e., \(\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots\)).
2. Transformations inside the Argument:
Consider the argument inside the secant function \(\left[2 \left( x - \frac{\pi}{2} \right)\right]\).
- The term \(2(x - \frac{\pi}{2})\) involves a horizontal shift and a horizontal scaling.
- The shift \(x - \frac{\pi}{2}\) shifts the function \(\sec(x)\) to the right by \(\frac{\pi}{2}\).
- The factor of 2 scales the function horizontally by a factor of \(\frac{1}{2}\), which means the period of the function is shortened by half.
3. Vertical Stretch and Shift:
The coefficient 3 outside the secant function, \(3 \sec(\cdots)\), stretches the graph vertically by a factor of 3.
The term \(+2\) outside the secant function shifts the graph upwards by 2 units.
### Key Features of the Graph:
- Period:
The basic period of \(\sec(x)\) is \(2\pi\). Since the argument has been scaled by 2 \(2 \left( x - \frac{\pi}{2} \right)\), the period of this secant function is \(\frac{2\pi}{2} = \pi\).
- Asymptotes:
The vertical asymptotes occur where \( \cos \left[2 \left( x - \frac{\pi}{2} \right)\right] = 0 \). Since \(\cos \theta = 0\) at \(\theta = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots\), we solve:
[tex]\[ 2 \left( x - \frac{\pi}{2} \right) = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots \][/tex]
Simplifying for \(x\), we get:
[tex]\[ x - \frac{\pi}{2} = \pm \frac{\pi}{4}, \pm \frac{3\pi}{4}, \ldots \][/tex]
[tex]\[ x = \frac{\pi}{2} \pm \frac{\pi}{4}, \frac{\pi}{2} \pm \frac{3\pi}{4}, \ldots \][/tex]
This simplifies to:
[tex]\[ x = \frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \ldots \][/tex]
- Behavior around Asymptotes:
Near each of these vertical asymptotes, the function will approach \(\pm \infty\).
- Midline and Amplitude:
The midline is \(y = 2\), and the maximum and minimum points (due to the vertical stretch) will oscillate around this line with an amplitude of 3.
Thus, the graph has vertical asymptotes at \(x = \frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \ldots\), a period of \(\pi\), and it oscillates with a vertical stretch by a factor of 3 and is shifted upward by 2 units.
Sketch the graph:
1. Draw the vertical asymptotes at \( x = \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \ldots\).
2. Plot the midline at \(y = 2\).
3. For \(y = 3 \sec \left[2 \left(x - \frac{\pi}{2}\right) \right] + 2\), around these asymptotes, the graph will have the shape similar to the sec function, stretched vertically by 3, oscillating around \( y = 2 \).
By following these steps, you can accurately draw or understand the graph of the function [tex]\(y = 3 \sec \left[2 \left(x - \frac{\pi}{2}\right) \right] + 2\)[/tex].
### Step-by-Step Solution:
1. Understanding the Secant Function:
The secant function, \(\sec(x)\), is the reciprocal of the cosine function, \(\cos(x)\). Thus, \(\sec(x) = \frac{1}{\cos(x)}\). The function \(\sec(x)\) has vertical asymptotes wherever \(\cos(x) = 0\), which occurs at odd multiples of \(\frac{\pi}{2}\) (i.e., \(\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots\)).
2. Transformations inside the Argument:
Consider the argument inside the secant function \(\left[2 \left( x - \frac{\pi}{2} \right)\right]\).
- The term \(2(x - \frac{\pi}{2})\) involves a horizontal shift and a horizontal scaling.
- The shift \(x - \frac{\pi}{2}\) shifts the function \(\sec(x)\) to the right by \(\frac{\pi}{2}\).
- The factor of 2 scales the function horizontally by a factor of \(\frac{1}{2}\), which means the period of the function is shortened by half.
3. Vertical Stretch and Shift:
The coefficient 3 outside the secant function, \(3 \sec(\cdots)\), stretches the graph vertically by a factor of 3.
The term \(+2\) outside the secant function shifts the graph upwards by 2 units.
### Key Features of the Graph:
- Period:
The basic period of \(\sec(x)\) is \(2\pi\). Since the argument has been scaled by 2 \(2 \left( x - \frac{\pi}{2} \right)\), the period of this secant function is \(\frac{2\pi}{2} = \pi\).
- Asymptotes:
The vertical asymptotes occur where \( \cos \left[2 \left( x - \frac{\pi}{2} \right)\right] = 0 \). Since \(\cos \theta = 0\) at \(\theta = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots\), we solve:
[tex]\[ 2 \left( x - \frac{\pi}{2} \right) = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots \][/tex]
Simplifying for \(x\), we get:
[tex]\[ x - \frac{\pi}{2} = \pm \frac{\pi}{4}, \pm \frac{3\pi}{4}, \ldots \][/tex]
[tex]\[ x = \frac{\pi}{2} \pm \frac{\pi}{4}, \frac{\pi}{2} \pm \frac{3\pi}{4}, \ldots \][/tex]
This simplifies to:
[tex]\[ x = \frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \ldots \][/tex]
- Behavior around Asymptotes:
Near each of these vertical asymptotes, the function will approach \(\pm \infty\).
- Midline and Amplitude:
The midline is \(y = 2\), and the maximum and minimum points (due to the vertical stretch) will oscillate around this line with an amplitude of 3.
Thus, the graph has vertical asymptotes at \(x = \frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \ldots\), a period of \(\pi\), and it oscillates with a vertical stretch by a factor of 3 and is shifted upward by 2 units.
Sketch the graph:
1. Draw the vertical asymptotes at \( x = \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \ldots\).
2. Plot the midline at \(y = 2\).
3. For \(y = 3 \sec \left[2 \left(x - \frac{\pi}{2}\right) \right] + 2\), around these asymptotes, the graph will have the shape similar to the sec function, stretched vertically by 3, oscillating around \( y = 2 \).
By following these steps, you can accurately draw or understand the graph of the function [tex]\(y = 3 \sec \left[2 \left(x - \frac{\pi}{2}\right) \right] + 2\)[/tex].
