At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Let's start by understanding the function [tex]\( f(x) = \sin(x) \)[/tex], which is the sine function. The sine function is a periodic function which oscillates between -1 and 1 with a period of [tex]\( 2\pi \)[/tex].
Now, consider adding a number [tex]\( n \)[/tex] to the function, resulting in [tex]\( f(x) = \sin(x) + n \)[/tex].
To analyze how the graph of this new function [tex]\( f(x) = \sin(x) + n \)[/tex] changes, let's examine each of the statements provided:
1. There is a vertical shift of n units.
- Adding [tex]\( n \)[/tex] to the function [tex]\( \sin(x) \)[/tex] will shift the entire graph vertically by [tex]\( n \)[/tex] units. If [tex]\( n \)[/tex] is positive, the graph shifts up by [tex]\( n \)[/tex] units. If [tex]\( n \)[/tex] is negative, the graph shifts down by [tex]\( n \)[/tex] units.
2. The x-intercepts will shift n units.
- The x-intercepts of [tex]\( \sin(x) \)[/tex] are the points where the function equals zero. When [tex]\( n \)[/tex] is added, the function becomes [tex]\( \sin(x) + n \)[/tex]. The new function [tex]\( \sin(x) + n \)[/tex] will equal zero when [tex]\( \sin(x) = -n \)[/tex]. The x-positions where this occurs generally will not be [tex]\( n \)[/tex] units shifted from the original x-intercepts. Thus, this statement is incorrect.
3. There is a change in amplitude of n units.
- The amplitude of [tex]\( \sin(x) \)[/tex] is the maximum absolute value it reaches, which is 1. Adding a constant [tex]\( n \)[/tex] does not change how much the function oscillates from its mean position; it merely shifts the graph up or down. Therefore, the amplitude remains 1, so this statement is incorrect.
4. The range will change by a factor of 2n units.
- The range of [tex]\( \sin(x) \)[/tex] is from -1 to 1. Adding [tex]\( n \)[/tex] will shift the entire range to [tex]\( n-1 \)[/tex] to [tex]\( n+1 \)[/tex]. The span of the range does not change; it is just shifted. The range doesn't increase or shrink by a factor of [tex]\( 2n \)[/tex] units. Therefore, this statement is incorrect.
Summarizing the correct statement:
- There is a vertical shift of n units. This accurately describes how adding a number [tex]\( n \)[/tex] to the function [tex]\( \sin(x) \)[/tex] affects its graph.
Answer: There is a vertical shift of n units.
Now, consider adding a number [tex]\( n \)[/tex] to the function, resulting in [tex]\( f(x) = \sin(x) + n \)[/tex].
To analyze how the graph of this new function [tex]\( f(x) = \sin(x) + n \)[/tex] changes, let's examine each of the statements provided:
1. There is a vertical shift of n units.
- Adding [tex]\( n \)[/tex] to the function [tex]\( \sin(x) \)[/tex] will shift the entire graph vertically by [tex]\( n \)[/tex] units. If [tex]\( n \)[/tex] is positive, the graph shifts up by [tex]\( n \)[/tex] units. If [tex]\( n \)[/tex] is negative, the graph shifts down by [tex]\( n \)[/tex] units.
2. The x-intercepts will shift n units.
- The x-intercepts of [tex]\( \sin(x) \)[/tex] are the points where the function equals zero. When [tex]\( n \)[/tex] is added, the function becomes [tex]\( \sin(x) + n \)[/tex]. The new function [tex]\( \sin(x) + n \)[/tex] will equal zero when [tex]\( \sin(x) = -n \)[/tex]. The x-positions where this occurs generally will not be [tex]\( n \)[/tex] units shifted from the original x-intercepts. Thus, this statement is incorrect.
3. There is a change in amplitude of n units.
- The amplitude of [tex]\( \sin(x) \)[/tex] is the maximum absolute value it reaches, which is 1. Adding a constant [tex]\( n \)[/tex] does not change how much the function oscillates from its mean position; it merely shifts the graph up or down. Therefore, the amplitude remains 1, so this statement is incorrect.
4. The range will change by a factor of 2n units.
- The range of [tex]\( \sin(x) \)[/tex] is from -1 to 1. Adding [tex]\( n \)[/tex] will shift the entire range to [tex]\( n-1 \)[/tex] to [tex]\( n+1 \)[/tex]. The span of the range does not change; it is just shifted. The range doesn't increase or shrink by a factor of [tex]\( 2n \)[/tex] units. Therefore, this statement is incorrect.
Summarizing the correct statement:
- There is a vertical shift of n units. This accurately describes how adding a number [tex]\( n \)[/tex] to the function [tex]\( \sin(x) \)[/tex] affects its graph.
Answer: There is a vertical shift of n units.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
