Certainly! Let's determine the amplitude and period of the function [tex]\( f(x) = -8 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) \)[/tex].
### Step 1: Simplify the Function Using Trigonometric Identities
First, we'll simplify the given function. We can use the double-angle identity for sine, which states:
[tex]\[ \sin(2a) = 2 \sin(a) \cos(a) \][/tex]
Let [tex]\( a = \frac{3x}{2} \)[/tex]. Then we have:
[tex]\[ \sin(2a) = \sin \left( 2 \cdot \frac{3x}{2} \right) = \sin(3x) \][/tex]
Thus,
[tex]\[ 2 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) = \sin(3x) \][/tex]
Given the function [tex]\( f(x) = -8 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) \)[/tex], we can rewrite it as:
[tex]\[ f(x) = -8 \left( \frac{1}{2} \sin(3x) \right) \][/tex]
[tex]\[ f(x) = -4 \sin(3x) \][/tex]
### Step 2: Determine the Amplitude
The amplitude of a sine or cosine function of the form [tex]\( y = A \sin(Bx) \)[/tex] or [tex]\( y = A \cos(Bx) \)[/tex] is the absolute value of the coefficient [tex]\( A \)[/tex].
For the function [tex]\( f(x) = -4 \sin(3x) \)[/tex]:
- The coefficient [tex]\( A \)[/tex] is [tex]\(-4\)[/tex].
- The amplitude is the absolute value of [tex]\( A \)[/tex], which is [tex]\( | -4 | = 4 \)[/tex].
### Step 3: Determine the Period
The period of a sine or cosine function [tex]\( y = \sin(Bx) \)[/tex] or [tex]\( y = \cos(Bx) \)[/tex] is given by [tex]\( \frac{2\pi}{B} \)[/tex].
For the function [tex]\( f(x) = -4 \sin(3x) \)[/tex]:
- The coefficient [tex]\( B \)[/tex] is [tex]\( 3 \)[/tex].
- Therefore, the period is [tex]\( \frac{2\pi}{3} \)[/tex].
### Summary
- Amplitude: [tex]\( 4 \)[/tex]
- Period: [tex]\( \frac{2\pi}{3} \approx 2.0943951023931953 \)[/tex]
Therefore, the amplitude and period of the graph of [tex]\( f(x) = -8 \sin \frac{3x}{2} \cos \frac{3x}{2} \)[/tex] are [tex]\( 4 \)[/tex] and [tex]\( \frac{2\pi}{3} \)[/tex], respectively.
