Answer :
Alright, let's analyze and describe the graph of the function [tex]\( f(x) = 4 \sin 2 \left( x - 45^\circ \right) + 2 \)[/tex] step-by-step:
1. Amplitude:
- The amplitude of the function can be observed directly from the coefficient in front of the sine function. Here, the coefficient is [tex]\(4\)[/tex]. Therefore, the amplitude is [tex]\(4\)[/tex].
2. Midline (Vertical Shift):
- The midline or vertical shift of the function is determined by the constant added outside the sine function. In this case, the function includes a [tex]\(+2\)[/tex]. Hence, the midline is [tex]\(y = 2\)[/tex].
3. Range:
- The range of the sine function is based on the amplitude and the vertical shift. The standard range of [tex]\(\sin(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. With an amplitude of [tex]\(4\)[/tex], the function [tex]\(\sin(x)\)[/tex] would range from [tex]\(-4\)[/tex] to [tex]\(4\)[/tex]. Adding the vertical shift [tex]\(+2\)[/tex], the range of the entire function [tex]\( f(x) \)[/tex] becomes:
[tex]\[ 2 - 4 = -2 \quad \text{(minimum value)} \][/tex]
[tex]\[ 2 + 4 = 6 \quad \text{(maximum value)} \][/tex]
So, the range of the function is from [tex]\(-2\)[/tex] to [tex]\(6\)[/tex], which can be written as [tex]\((-2, 6)\)[/tex].
4. Period:
- The period of a sine function [tex]\( a \sin(bx) \)[/tex] is given by [tex]\(\frac{2\pi}{b}\)[/tex]. In this function, the value of [tex]\(b\)[/tex] is [tex]\(2\)[/tex]. Thus, the period of the function is:
[tex]\[ \frac{2\pi}{2} = \pi \][/tex]
So, the period of the function [tex]\( f(x) \)[/tex] is [tex]\(\pi\)[/tex].
5. Horizontal Translation (Phase Shift):
- The horizontal translation or phase shift of the sine function is given by the term inside the sine function [tex]\((x - \text{horizontal shift})\)[/tex]. In this case, the horizontal shift is [tex]\(45^\circ\)[/tex]. To convert degrees to radians (since we're typically focusing on radians in trigonometric functions):
[tex]\[ 45^\circ = \frac{\pi}{4} \, \text{radians} \][/tex]
Therefore, the function is horizontally shifted to the right by [tex]\(\frac{\pi}{4}\)[/tex] radians.
### Description of the Graph:
- Amplitude: The graph oscillates 4 units above and below its midline.
- Midline (Vertical Shift): The central axis or baseline of the graph is [tex]\(y = 2\)[/tex].
- Range: The values of [tex]\(f(x)\)[/tex] will lie between [tex]\(-2\)[/tex] and [tex]\(6\)[/tex].
- Period: The function completes one full oscillation over an interval of [tex]\(\pi\)[/tex] units.
- Horizontal Translation: The graph is shifted to the right by [tex]\(\frac{\pi}{4}\)[/tex] radians.
### Sketch:
1. Start by drawing the midline at [tex]\(y = 2\)[/tex].
2. Indicate the amplitude: The graph will reach up to [tex]\(y = 6\)[/tex] (midline [tex]\(+4\)[/tex]) and down to [tex]\(y = -2\)[/tex] (midline [tex]\(-4\)[/tex]).
3. Draw a sinusoidal wave starting from [tex]\(x = \frac{\pi}{4}\)[/tex] (due to the phase shift).
4. Complete the wave cycle over the interval [tex]\([ \frac{\pi}{4}, \frac{\pi}{4} + \pi ]\)[/tex].
5. Ensure the wave oscillates 4 units above and below the midline within each period.
In this manner, you can sketch and describe the comprehensive behavior and appearance of the function [tex]\( f(x) = 4 \sin 2 \left( x - 45^\circ \right) + 2 \)[/tex].
1. Amplitude:
- The amplitude of the function can be observed directly from the coefficient in front of the sine function. Here, the coefficient is [tex]\(4\)[/tex]. Therefore, the amplitude is [tex]\(4\)[/tex].
2. Midline (Vertical Shift):
- The midline or vertical shift of the function is determined by the constant added outside the sine function. In this case, the function includes a [tex]\(+2\)[/tex]. Hence, the midline is [tex]\(y = 2\)[/tex].
3. Range:
- The range of the sine function is based on the amplitude and the vertical shift. The standard range of [tex]\(\sin(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. With an amplitude of [tex]\(4\)[/tex], the function [tex]\(\sin(x)\)[/tex] would range from [tex]\(-4\)[/tex] to [tex]\(4\)[/tex]. Adding the vertical shift [tex]\(+2\)[/tex], the range of the entire function [tex]\( f(x) \)[/tex] becomes:
[tex]\[ 2 - 4 = -2 \quad \text{(minimum value)} \][/tex]
[tex]\[ 2 + 4 = 6 \quad \text{(maximum value)} \][/tex]
So, the range of the function is from [tex]\(-2\)[/tex] to [tex]\(6\)[/tex], which can be written as [tex]\((-2, 6)\)[/tex].
4. Period:
- The period of a sine function [tex]\( a \sin(bx) \)[/tex] is given by [tex]\(\frac{2\pi}{b}\)[/tex]. In this function, the value of [tex]\(b\)[/tex] is [tex]\(2\)[/tex]. Thus, the period of the function is:
[tex]\[ \frac{2\pi}{2} = \pi \][/tex]
So, the period of the function [tex]\( f(x) \)[/tex] is [tex]\(\pi\)[/tex].
5. Horizontal Translation (Phase Shift):
- The horizontal translation or phase shift of the sine function is given by the term inside the sine function [tex]\((x - \text{horizontal shift})\)[/tex]. In this case, the horizontal shift is [tex]\(45^\circ\)[/tex]. To convert degrees to radians (since we're typically focusing on radians in trigonometric functions):
[tex]\[ 45^\circ = \frac{\pi}{4} \, \text{radians} \][/tex]
Therefore, the function is horizontally shifted to the right by [tex]\(\frac{\pi}{4}\)[/tex] radians.
### Description of the Graph:
- Amplitude: The graph oscillates 4 units above and below its midline.
- Midline (Vertical Shift): The central axis or baseline of the graph is [tex]\(y = 2\)[/tex].
- Range: The values of [tex]\(f(x)\)[/tex] will lie between [tex]\(-2\)[/tex] and [tex]\(6\)[/tex].
- Period: The function completes one full oscillation over an interval of [tex]\(\pi\)[/tex] units.
- Horizontal Translation: The graph is shifted to the right by [tex]\(\frac{\pi}{4}\)[/tex] radians.
### Sketch:
1. Start by drawing the midline at [tex]\(y = 2\)[/tex].
2. Indicate the amplitude: The graph will reach up to [tex]\(y = 6\)[/tex] (midline [tex]\(+4\)[/tex]) and down to [tex]\(y = -2\)[/tex] (midline [tex]\(-4\)[/tex]).
3. Draw a sinusoidal wave starting from [tex]\(x = \frac{\pi}{4}\)[/tex] (due to the phase shift).
4. Complete the wave cycle over the interval [tex]\([ \frac{\pi}{4}, \frac{\pi}{4} + \pi ]\)[/tex].
5. Ensure the wave oscillates 4 units above and below the midline within each period.
In this manner, you can sketch and describe the comprehensive behavior and appearance of the function [tex]\( f(x) = 4 \sin 2 \left( x - 45^\circ \right) + 2 \)[/tex].
