Answer :
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.
