Sure, let's break down the characteristics of the function [tex]\( y = -3 \sin (2\pi x - 4) \)[/tex] step by step:
1. Amplitude:
The amplitude of a sine function [tex]\( y = a \sin(bx - c) + d \)[/tex] is given by the absolute value of the coefficient [tex]\( a \)[/tex]. In this case, [tex]\( a = -3 \)[/tex], so the amplitude is:
[tex]\[
\text{Amplitude} = |a| = |-3| = 3
\][/tex]
2. Period:
The period of a sine function is determined by the coefficient [tex]\( b \)[/tex] inside the function. The period [tex]\( T \)[/tex] is given by the formula:
[tex]\[
\text{Period} = \frac{2\pi}{|b|}
\][/tex]
For the given function, [tex]\( b = 2\pi \)[/tex]. Therefore:
[tex]\[
\text{Period} = \frac{2\pi}{2\pi} = 1
\][/tex]
3. Horizontal Shift:
The horizontal shift (or phase shift) of the sine function is found by solving the expression [tex]\(\frac{c}{b}\)[/tex], where [tex]\( c \)[/tex] is the phase shift inside the function. For our function:
[tex]\[
c = 4 \quad \text{and} \quad b = 2\pi
\][/tex]
Thus, the horizontal shift is:
[tex]\[
\text{Horizontal Shift} = \frac{c}{b} = \frac{4}{2\pi} = \frac{4}{2\pi} \approx 0.6366
\][/tex]
So, the correct answers are:
- Amplitude: [tex]\( 3 \)[/tex]
- Period: [tex]\( 1.0 \)[/tex]
- Horizontal shift: [tex]\( 0.6366 \)[/tex]
