To analyze the functions \( f(x) = \sin(x) \) and \( g(x) = \sin(2x) + 3 \), let's examine their properties:
-
Amplitude:
- The amplitude of \( f(x) = \sin(x) \) is \( 1 \) (since the sine function oscillates between -1 and 1).
- The amplitude of \( g(x) = \sin(2x) + 3 \) is also \( 1 \). The amplitude remains unchanged regardless of vertical shifts.
- Therefore, the amplitude of \( g(x) \) is NOT double the amplitude of \( f(x) \).
-
Zeroes:
- The zeroes of \( f(x) = \sin(x) \) are at \( n\pi \) for \( n \in \mathbb{Z} \).
- To find the zeroes of \( g(x) = \sin(2x) + 3 \), we set it to zero: \( \sin(2x) + 3 = 0\) implies \( \sin(2x) = -3 \), which is not possible since the sine function can only output values in the range [-1, 1]. Thus, \( g(x) \) has no zeroes.
- Therefore, \( g(x) \) and \( f(x) \) do NOT have the same zeroes.
-
Vertical Shift:
- The function \( g(x) = \sin(2x) + 3 \) is shifted 3 units upward compared to \( f(x) = \sin(x) \).
- Therefore, \( g(x) \) is shifted 3 units higher than \( f(x) \).
-
Horizontal Shift:
- Neither \( g(x) \) nor \( f(x) \) exhibits a horizontal shift. Instead, \( g(x) \) includes a change in the frequency (due to the factor of 2 in \( \sin(2x) \)) which does not equate to a horizontal shift.
-
Period:
- The period of \( f(x) = \sin(x) \) is \( 2\pi \).
- The period of \( g(x) = \sin(2x) + 3 \) is \( \frac{2\pi}{2} = \pi \).
- Therefore, the period of \( g(x) \) is half the period of \( f(x) \).
Based on this analysis, the correct responses are:
- g(x) is shifted 3 units higher than f(x).
- g(x) has a period that is half the period of f(x).
These are the applicable comparisons between the functions \( f(x) \) and \( g(x) \).
1 answer