Find the period and the amplitude.
y=-cos (8x/17)
STEP 1: Rewrite the trigonometric expression in the form ya cos (bx).
y=-cos (8x/17) -
cos(
STEP 2: Find the period of y by dividing 2 by b.
x)
Does this period represent a horizontal stretching of the graph of y = a cos (x) or a horizontal shrinking of the graph of y = a cos (x)?
O horizontal stretching
O horizontal shrinking
STEP 3: Will the amplitude of y be positive or negative?
O Positive, because both 8 and 17 are positive.
O Positive, because the leading coefficient of y is negative.
Negative, because both 8 and 17 are positive.
O Positive, because amplitude measures a distance.
O Negative, because the leading coefficient of y is negative.
Find the amplitude of y.



Answer :

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]!