Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To sketch the graphs of [tex]\( f(x) = 2 \sin x - 3 \)[/tex] and [tex]\( g(x) = -2 \cos x + 4 \)[/tex] for [tex]\( x \in [90^\circ, 270^\circ] \)[/tex], let's follow these steps:
### 1. Understanding the Basic Functions
First, note the basic forms of the sine and cosine functions within the interval.
- [tex]\( \sin x \)[/tex] ranges from 1 at [tex]\( x = 90^\circ \)[/tex] to -1 at [tex]\( x = 270^\circ \)[/tex].
- [tex]\( \cos x \)[/tex] ranges from 0 at [tex]\( x = 90^\circ \)[/tex] to 0 again at [tex]\( x = 270^\circ \)[/tex], but it achieves -1 at [tex]\( x = 180^\circ \)[/tex].
### 2. Apply Transformations to the Sine and Cosine Functions
Next, apply the transformations to the sine and cosine functions.
For [tex]\( f(x) = 2 \sin x - 3 \)[/tex]:
- Multiply the sine function by 2: [tex]\( 2 \sin x \)[/tex]. This scales the amplitude to range from -2 to 2.
- Subtract 3: [tex]\( 2 \sin x - 3 \)[/tex]. This shifts the entire graph downward by 3 units.
So, for [tex]\( x \)[/tex] in [tex]\([90^\circ, 270^\circ]\)[/tex],
[tex]\[ \begin{align*} f(90^\circ) & = 2 \sin 90^\circ - 3 = 2 \cdot 1 - 3 = 2 - 3 = -1, \\ f(180^\circ) & = 2 \sin 180^\circ - 3 = 2 \cdot 0 - 3 = -3, \\ f(270^\circ) & = 2 \sin 270^\circ - 3 = 2 \cdot (-1) - 3 = -2 - 3 = -5. \end{align*} \][/tex]
For [tex]\( g(x) = -2 \cos x + 4 \)[/tex]:
- Multiply the cosine function by -2: [tex]\( -2 \cos x \)[/tex]. This scales and reflects the cosine function.
- Add 4: [tex]\( -2 \cos x + 4 \)[/tex]. This shifts the graph upward by 4 units.
So, for [tex]\( x \)[/tex] in [tex]\([90^\circ, 270^\circ]\)[/tex],
[tex]\[ \begin{align*} g(90^\circ) & = -2 \cos 90^\circ + 4 = -2 \cdot 0 + 4 = 4, \\ g(180^\circ) & = -2 \cos 180^\circ + 4 = -2 \cdot (-1) + 4 = 2 + 4 = 6, \\ g(270^\circ) & = -2 \cos 270^\circ + 4 = -2 \cdot 0 + 4 = 4. \end{align*} \][/tex]
### 3. Graph the Functions
Using the calculated points, you can start sketching the graphs.
#### [tex]\( f(x) = 2 \sin x - 3 \)[/tex]:
- At [tex]\( x = 90^\circ \)[/tex], [tex]\( f(x) = -1 \)[/tex]
- At [tex]\( x = 180^\circ \)[/tex], [tex]\( f(x) = -3 \)[/tex]
- At [tex]\( x = 270^\circ \)[/tex], [tex]\( f(x) = -5 \)[/tex]
Moreover, between these points, [tex]\( f(x) \)[/tex] shows a sinusoidal behavior.
#### [tex]\( g(x) = -2 \cos x + 4 \)[/tex]:
- At [tex]\( x = 90^\circ \)[/tex], [tex]\( g(x) = 4 \)[/tex]
- At [tex]\( x = 180^\circ \)[/tex], [tex]\( g(x) = 6 \)[/tex]
- At [tex]\( x = 270^\circ \)[/tex], [tex]\( g(x) = 4 \)[/tex]
Similarly, between these points, [tex]\( g(x) \)[/tex] exhibits a cosine behavior but reflected and shifted.
### 4. Plotting the Graphs
1. Start with the [tex]\( x \)[/tex]-axis ranging from [tex]\( 90^\circ \)[/tex] to [tex]\( 270^\circ \)[/tex].
2. [tex]\( f(x) \)[/tex] will start from [tex]\( (90^\circ, -1) \)[/tex], dip to [tex]\( (180^\circ, -3) \)[/tex], and further to [tex]\( (270^\circ, -5) \)[/tex].
3. [tex]\( g(x) \)[/tex] starts from [tex]\( (90^\circ, 4) \)[/tex], rises to [tex]\( (180^\circ, 6) \)[/tex], and returns to [tex]\( (270^\circ, 4) \)[/tex].
The sinusoidal curve of [tex]\( f(x) \)[/tex] will be downward shifted, while [tex]\( g(x) \)[/tex] will be an inverted cosine curve shifted upwards.
### Graph Sketch
Here's a conceptual sketch of the graphs:
[tex]\[ \begin{array}{c} x \quad \quad f(x) \quad \quad g(x) \\ \hline 90^\circ \, | \, -1 \quad \quad 4 \\ 135^\circ \, | \, -2 \quad \quad 5 \\ 180^\circ \, | \, -3 \quad \quad 6 \\ 225^\circ \, | \, -4 \quad \quad 5 \\ 270^\circ \, | \, -5 \quad \quad 4 \end{array} \][/tex]
Plot the above points and draw smooth curves through them to complete the graphs.
### 1. Understanding the Basic Functions
First, note the basic forms of the sine and cosine functions within the interval.
- [tex]\( \sin x \)[/tex] ranges from 1 at [tex]\( x = 90^\circ \)[/tex] to -1 at [tex]\( x = 270^\circ \)[/tex].
- [tex]\( \cos x \)[/tex] ranges from 0 at [tex]\( x = 90^\circ \)[/tex] to 0 again at [tex]\( x = 270^\circ \)[/tex], but it achieves -1 at [tex]\( x = 180^\circ \)[/tex].
### 2. Apply Transformations to the Sine and Cosine Functions
Next, apply the transformations to the sine and cosine functions.
For [tex]\( f(x) = 2 \sin x - 3 \)[/tex]:
- Multiply the sine function by 2: [tex]\( 2 \sin x \)[/tex]. This scales the amplitude to range from -2 to 2.
- Subtract 3: [tex]\( 2 \sin x - 3 \)[/tex]. This shifts the entire graph downward by 3 units.
So, for [tex]\( x \)[/tex] in [tex]\([90^\circ, 270^\circ]\)[/tex],
[tex]\[ \begin{align*} f(90^\circ) & = 2 \sin 90^\circ - 3 = 2 \cdot 1 - 3 = 2 - 3 = -1, \\ f(180^\circ) & = 2 \sin 180^\circ - 3 = 2 \cdot 0 - 3 = -3, \\ f(270^\circ) & = 2 \sin 270^\circ - 3 = 2 \cdot (-1) - 3 = -2 - 3 = -5. \end{align*} \][/tex]
For [tex]\( g(x) = -2 \cos x + 4 \)[/tex]:
- Multiply the cosine function by -2: [tex]\( -2 \cos x \)[/tex]. This scales and reflects the cosine function.
- Add 4: [tex]\( -2 \cos x + 4 \)[/tex]. This shifts the graph upward by 4 units.
So, for [tex]\( x \)[/tex] in [tex]\([90^\circ, 270^\circ]\)[/tex],
[tex]\[ \begin{align*} g(90^\circ) & = -2 \cos 90^\circ + 4 = -2 \cdot 0 + 4 = 4, \\ g(180^\circ) & = -2 \cos 180^\circ + 4 = -2 \cdot (-1) + 4 = 2 + 4 = 6, \\ g(270^\circ) & = -2 \cos 270^\circ + 4 = -2 \cdot 0 + 4 = 4. \end{align*} \][/tex]
### 3. Graph the Functions
Using the calculated points, you can start sketching the graphs.
#### [tex]\( f(x) = 2 \sin x - 3 \)[/tex]:
- At [tex]\( x = 90^\circ \)[/tex], [tex]\( f(x) = -1 \)[/tex]
- At [tex]\( x = 180^\circ \)[/tex], [tex]\( f(x) = -3 \)[/tex]
- At [tex]\( x = 270^\circ \)[/tex], [tex]\( f(x) = -5 \)[/tex]
Moreover, between these points, [tex]\( f(x) \)[/tex] shows a sinusoidal behavior.
#### [tex]\( g(x) = -2 \cos x + 4 \)[/tex]:
- At [tex]\( x = 90^\circ \)[/tex], [tex]\( g(x) = 4 \)[/tex]
- At [tex]\( x = 180^\circ \)[/tex], [tex]\( g(x) = 6 \)[/tex]
- At [tex]\( x = 270^\circ \)[/tex], [tex]\( g(x) = 4 \)[/tex]
Similarly, between these points, [tex]\( g(x) \)[/tex] exhibits a cosine behavior but reflected and shifted.
### 4. Plotting the Graphs
1. Start with the [tex]\( x \)[/tex]-axis ranging from [tex]\( 90^\circ \)[/tex] to [tex]\( 270^\circ \)[/tex].
2. [tex]\( f(x) \)[/tex] will start from [tex]\( (90^\circ, -1) \)[/tex], dip to [tex]\( (180^\circ, -3) \)[/tex], and further to [tex]\( (270^\circ, -5) \)[/tex].
3. [tex]\( g(x) \)[/tex] starts from [tex]\( (90^\circ, 4) \)[/tex], rises to [tex]\( (180^\circ, 6) \)[/tex], and returns to [tex]\( (270^\circ, 4) \)[/tex].
The sinusoidal curve of [tex]\( f(x) \)[/tex] will be downward shifted, while [tex]\( g(x) \)[/tex] will be an inverted cosine curve shifted upwards.
### Graph Sketch
Here's a conceptual sketch of the graphs:
[tex]\[ \begin{array}{c} x \quad \quad f(x) \quad \quad g(x) \\ \hline 90^\circ \, | \, -1 \quad \quad 4 \\ 135^\circ \, | \, -2 \quad \quad 5 \\ 180^\circ \, | \, -3 \quad \quad 6 \\ 225^\circ \, | \, -4 \quad \quad 5 \\ 270^\circ \, | \, -5 \quad \quad 4 \end{array} \][/tex]
Plot the above points and draw smooth curves through them to complete the graphs.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
