At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
