Certainly! Let's go through the solution step-by-step for the given trigonometric expression [tex]\( y = -\cos\left(\frac{8x}{17}\right) \)[/tex].
### STEP 1: Rewrite the Trigonometric Expression
We start with the expression:
[tex]\[ y = -\cos\left(\frac{8x}{17}\right) \][/tex]
To make it easier to analyze, we'll rewrite it in the standard form [tex]\( y = a \cos(bx) \)[/tex]:
- In this expression, [tex]\( a = -1 \)[/tex]
- And [tex]\( b = \frac{8}{17} \)[/tex]
### STEP 2: Find the Period of [tex]\( y \)[/tex]
The period of the function [tex]\( y = \cos(bx) \)[/tex] is given by:
[tex]\[ \text{Period} = \frac{2\pi}{b} \][/tex]
Plugging in the value of [tex]\( b \)[/tex]:
[tex]\[ \text{Period} = \frac{2\pi}{\frac{8}{17}} = \frac{2\pi \times 17}{8} \approx 13.351768777756622 \][/tex]
#### Determine Horizontal Change
We must determine if this represents horizontal stretching or shrinking. Specifically:
- For [tex]\( b < 1 \)[/tex], the graph undergoes horizontal stretching.
- For [tex]\( b > 1 \)[/tex], the graph undergoes horizontal shrinking.
Since [tex]\( b = \frac{8}{17} \approx 0.47 < 1 \)[/tex], it represents a horizontal stretching of the graph.
### STEP 3: Find the Amplitude of [tex]\( y \)[/tex]
The amplitude is the absolute value of the coefficient [tex]\( a \)[/tex] (which is always positive because amplitude measures the distance from the maximum to the baseline).
For our function:
[tex]\[ a = -1 \][/tex]
Thus, the amplitude is:
[tex]\[ \text{Amplitude} = |a| = 1 \][/tex]
#### Amplitude Sign Description
The amplitude will always be positive because amplitude measures a distance.
### Summary
- Period: [tex]\( 13.351768777756622 \)[/tex]
- Horizontal Change: Horizontal stretching
- Amplitude: [tex]\( 1 \)[/tex]
- Amplitude Sign Description: Positive, because amplitude measures a distance.
I hope this helps clarify the steps needed to analyze the trigonometric function [tex]\( y = -\cos\left(\frac{8x}{17}\right) \)[/tex]!
