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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.
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