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Sagot :
To graph the function [tex]\( y = \frac{-5}{2} \sec \left( \frac{1}{2} x - \frac{3\pi}{4} \right) \)[/tex], we need to determine the transformations of the basic secant function [tex]\( y = \sec(x) \)[/tex].
### Step-by-Step Solution
#### 1. Reflection across the [tex]\( x \)[/tex]-axis
- The coefficient [tex]\(\frac{-5}{2}\)[/tex] introduces a reflection across the [tex]\( x \)[/tex]-axis because it is negative. This means that any positive values of [tex]\( y \)[/tex] in [tex]\( \sec(x) \)[/tex] will become negative, and vice versa.
Answer: Reflect graph across [tex]\( x \)[/tex]-axis
#### 2. Stretching/Compressing Vertically
- The coefficient [tex]\(\frac{-5}{2}\)[/tex] also affects the vertical stretch/compression.
- Since [tex]\(\frac{-5}{2}\)[/tex] is larger than 1 in absolute value, it will stretch the graph vertically by a factor of [tex]\(\frac{5}{2}\)[/tex].
Answer: Stretch graph vertically
#### 3. Horizontal Stretch (Period change)
- The argument of the secant function, [tex]\(\frac{1}{2} x - \frac{3\pi}{4}\)[/tex], affects the period.
- The period of the basic secant function [tex]\( \sec(x) \)[/tex] is [tex]\( 2\pi \)[/tex].
- For [tex]\( y = \sec \left( \frac{1}{2} x \right) \)[/tex], the period will be [tex]\(\frac{2\pi}{\frac{1}{2}} = 4\pi\)[/tex]. This means the graph will be stretched horizontally.
Answer: Stretch graph horizontally
#### 4. Phase Shift (Horizontal Shift)
- The term [tex]\(- \frac{3\pi}{4}\)[/tex] inside the argument [tex]\(\frac{1}{2} x - \frac{3\pi}{4}\)[/tex] indicates a horizontal shift (phase shift).
- To find the shift, solve for [tex]\(x\)[/tex] in [tex]\(\frac{1}{2} x - \frac{3\pi}{4} = 0\)[/tex]:
[tex]\[ \frac{1}{2} x = \frac{3\pi}{4} \][/tex]
[tex]\[ x = 3\pi \cdot 2 = \frac{3\pi}{2} \][/tex]
- Therefore, the graph is shifted to the right by [tex]\(\frac{3\pi}{2}\)[/tex].
Answer: Shift graph horizontally to the right
### Summary of Transformations
1. Reflect across the [tex]\( x \)[/tex]-axis
2. Stretch vertically by a factor of [tex]\(\frac{5}{2}\)[/tex]
3. Stretch horizontally (period change to [tex]\( 4\pi \)[/tex])
4. Shift horizontally to the right by [tex]\(\frac{3\pi}{2}\)[/tex]
By applying these transformations, you should be able to graph the function [tex]\( y = \frac{-5}{2} \sec \left( \frac{1}{2} x - \frac{3\pi}{4} \right) \)[/tex]. The key features to plot include the location of vertical asymptotes (which will occur at the points where [tex]\( \cos \left( \frac{1}{2} x - \frac{3\pi}{4} \right) = 0 \)[/tex]), and the behavior near these asymptotes.
To graph:
- Draw the vertical asymptotes.
- Reflect the basic shape across the [tex]\( x \)[/tex]-axis.
- Stretch the graph vertically by a factor of [tex]\(\frac{5}{2}\)[/tex] (values will be [tex]\( \pm \frac{5}{2} \)[/tex]).
- Apply the horizontal stretching to change the period to [tex]\( 4\pi \)[/tex].
- Finally, shift the entire graph to the right by [tex]\(\frac{3\pi}{2}\)[/tex].
You should plot a few key points, and sketch the general behavior near these vertical asymptotes to complete the graph.
### Step-by-Step Solution
#### 1. Reflection across the [tex]\( x \)[/tex]-axis
- The coefficient [tex]\(\frac{-5}{2}\)[/tex] introduces a reflection across the [tex]\( x \)[/tex]-axis because it is negative. This means that any positive values of [tex]\( y \)[/tex] in [tex]\( \sec(x) \)[/tex] will become negative, and vice versa.
Answer: Reflect graph across [tex]\( x \)[/tex]-axis
#### 2. Stretching/Compressing Vertically
- The coefficient [tex]\(\frac{-5}{2}\)[/tex] also affects the vertical stretch/compression.
- Since [tex]\(\frac{-5}{2}\)[/tex] is larger than 1 in absolute value, it will stretch the graph vertically by a factor of [tex]\(\frac{5}{2}\)[/tex].
Answer: Stretch graph vertically
#### 3. Horizontal Stretch (Period change)
- The argument of the secant function, [tex]\(\frac{1}{2} x - \frac{3\pi}{4}\)[/tex], affects the period.
- The period of the basic secant function [tex]\( \sec(x) \)[/tex] is [tex]\( 2\pi \)[/tex].
- For [tex]\( y = \sec \left( \frac{1}{2} x \right) \)[/tex], the period will be [tex]\(\frac{2\pi}{\frac{1}{2}} = 4\pi\)[/tex]. This means the graph will be stretched horizontally.
Answer: Stretch graph horizontally
#### 4. Phase Shift (Horizontal Shift)
- The term [tex]\(- \frac{3\pi}{4}\)[/tex] inside the argument [tex]\(\frac{1}{2} x - \frac{3\pi}{4}\)[/tex] indicates a horizontal shift (phase shift).
- To find the shift, solve for [tex]\(x\)[/tex] in [tex]\(\frac{1}{2} x - \frac{3\pi}{4} = 0\)[/tex]:
[tex]\[ \frac{1}{2} x = \frac{3\pi}{4} \][/tex]
[tex]\[ x = 3\pi \cdot 2 = \frac{3\pi}{2} \][/tex]
- Therefore, the graph is shifted to the right by [tex]\(\frac{3\pi}{2}\)[/tex].
Answer: Shift graph horizontally to the right
### Summary of Transformations
1. Reflect across the [tex]\( x \)[/tex]-axis
2. Stretch vertically by a factor of [tex]\(\frac{5}{2}\)[/tex]
3. Stretch horizontally (period change to [tex]\( 4\pi \)[/tex])
4. Shift horizontally to the right by [tex]\(\frac{3\pi}{2}\)[/tex]
By applying these transformations, you should be able to graph the function [tex]\( y = \frac{-5}{2} \sec \left( \frac{1}{2} x - \frac{3\pi}{4} \right) \)[/tex]. The key features to plot include the location of vertical asymptotes (which will occur at the points where [tex]\( \cos \left( \frac{1}{2} x - \frac{3\pi}{4} \right) = 0 \)[/tex]), and the behavior near these asymptotes.
To graph:
- Draw the vertical asymptotes.
- Reflect the basic shape across the [tex]\( x \)[/tex]-axis.
- Stretch the graph vertically by a factor of [tex]\(\frac{5}{2}\)[/tex] (values will be [tex]\( \pm \frac{5}{2} \)[/tex]).
- Apply the horizontal stretching to change the period to [tex]\( 4\pi \)[/tex].
- Finally, shift the entire graph to the right by [tex]\(\frac{3\pi}{2}\)[/tex].
You should plot a few key points, and sketch the general behavior near these vertical asymptotes to complete the graph.
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