Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine the graph of the function [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex], we need to perform a series of transformations on the basic graph of the tangent function, [tex]\( y = \tan(x) \)[/tex]. Here's a step-by-step explanation of how to transform the graph:
1. Horizontal Shift:
The expression inside the tangent function is [tex]\( x + \frac{3\pi}{4} \)[/tex]. This represents a horizontal shift. Specifically, [tex]\( \tan(x) \)[/tex] shifts to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
2. Vertical Stretch:
The factor of 2 in front of the tangent function [tex]\( y = 2 \tan(x) \)[/tex] signifies a vertical stretch. Thus, every point on the graph of [tex]\( \tan(x) \)[/tex] is stretched vertically by a factor of 2. This means that the y-values are twice as far from the x-axis compared with the graph of [tex]\( y = \tan(x) \)[/tex].
### Step-by-Step Transformation:
1. Start with the basic graph of [tex]\( y = \tan(x) \)[/tex]:
- The graph has vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], where [tex]\( k \)[/tex] is any integer.
- The function [tex]\( \tan(x) \)[/tex] has a period of [tex]\( \pi \)[/tex], repeating every [tex]\( \pi \)[/tex] units.
- The x-intercepts are at [tex]\( x = k\pi \)[/tex].
2. Apply the horizontal shift by [tex]\( \frac{3\pi}{4} \)[/tex] units to the left:
- The vertical asymptotes of the new function [tex]\( y = \tan \left( x + \frac{3\pi}{4} \right) \)[/tex] will now occur at [tex]\( x = -\frac{3\pi}{4} + \frac{\pi}{2} + k\pi = \frac{\pi}{4} + k\pi - \frac{3\pi}{4} \)[/tex].
- Simplifying, the vertical asymptotes will be at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex].
- The x-intercepts of the function will be shifted to [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex].
3. Apply the vertical stretch by a factor of 2:
- The points on the original graph of [tex]\( \tan(x) \)[/tex] that were at [tex]\( (x, y) \)[/tex] become [tex]\( (x, 2y) \)[/tex].
### Characteristics of the new function [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex]:
- Period: The period remains [tex]\( \pi \)[/tex].
- Vertical Stretch Factor: The function now has a vertical stretch factor of 2.
- Vertical Asymptotes: They are at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex], for any integer [tex]\( k \)[/tex].
- X-intercepts: They are at [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex].
The graph has the same general shape as the tangent function, with vertical asymptotes, repeating every [tex]\( \pi \)[/tex] units, but it has been shifted and stretched. The graph will look like [tex]\( y = \tan(x) \)[/tex], but shifted left by [tex]\( \frac{3\pi}{4} \)[/tex] and stretched vertically by a factor of 2.
1. Horizontal Shift:
The expression inside the tangent function is [tex]\( x + \frac{3\pi}{4} \)[/tex]. This represents a horizontal shift. Specifically, [tex]\( \tan(x) \)[/tex] shifts to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
2. Vertical Stretch:
The factor of 2 in front of the tangent function [tex]\( y = 2 \tan(x) \)[/tex] signifies a vertical stretch. Thus, every point on the graph of [tex]\( \tan(x) \)[/tex] is stretched vertically by a factor of 2. This means that the y-values are twice as far from the x-axis compared with the graph of [tex]\( y = \tan(x) \)[/tex].
### Step-by-Step Transformation:
1. Start with the basic graph of [tex]\( y = \tan(x) \)[/tex]:
- The graph has vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], where [tex]\( k \)[/tex] is any integer.
- The function [tex]\( \tan(x) \)[/tex] has a period of [tex]\( \pi \)[/tex], repeating every [tex]\( \pi \)[/tex] units.
- The x-intercepts are at [tex]\( x = k\pi \)[/tex].
2. Apply the horizontal shift by [tex]\( \frac{3\pi}{4} \)[/tex] units to the left:
- The vertical asymptotes of the new function [tex]\( y = \tan \left( x + \frac{3\pi}{4} \right) \)[/tex] will now occur at [tex]\( x = -\frac{3\pi}{4} + \frac{\pi}{2} + k\pi = \frac{\pi}{4} + k\pi - \frac{3\pi}{4} \)[/tex].
- Simplifying, the vertical asymptotes will be at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex].
- The x-intercepts of the function will be shifted to [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex].
3. Apply the vertical stretch by a factor of 2:
- The points on the original graph of [tex]\( \tan(x) \)[/tex] that were at [tex]\( (x, y) \)[/tex] become [tex]\( (x, 2y) \)[/tex].
### Characteristics of the new function [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex]:
- Period: The period remains [tex]\( \pi \)[/tex].
- Vertical Stretch Factor: The function now has a vertical stretch factor of 2.
- Vertical Asymptotes: They are at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex], for any integer [tex]\( k \)[/tex].
- X-intercepts: They are at [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex].
The graph has the same general shape as the tangent function, with vertical asymptotes, repeating every [tex]\( \pi \)[/tex] units, but it has been shifted and stretched. The graph will look like [tex]\( y = \tan(x) \)[/tex], but shifted left by [tex]\( \frac{3\pi}{4} \)[/tex] and stretched vertically by a factor of 2.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
