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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.

Verified Answer :

GinnyAnswer

Certainly! Let's go through the solution step-by-step for the given trigonometric expression \(\( y = -\cos\left(\frac{8x}{17}\right) \)\).

### STEP 1: Rewrite the Trigonometric Expression
We start with the expression:
\(\[ y = -\cos\left(\frac{8x}{17}\right) \]\)

To make it easier to analyze, we'll rewrite it in the standard form \(\( y = a \cos(bx) \)\):
- In this expression, \(\( a = -1 \)\)
- And \(\( b = \frac{8}{17} \)\)

### STEP 2: Find the Period of \(\( y \)\)
The period of the function \(\( y = \cos(bx) \)\) is given by:
\(\[ \text{Period} = \frac{2\pi}{b} \]\)

Plugging in the value of \(\( b \)\):
\(\[ \text{Period} = \frac{2\pi}{\frac{8}{17}} = \frac{2\pi \times 17}{8} \approx 13.351768777756622 \]\)

#### Determine Horizontal Change
We must determine if this represents horizontal stretching or shrinking. Specifically:
- For \(\( b < 1 \)\), the graph undergoes horizontal stretching.
- For \(\( b > 1 \)\), the graph undergoes horizontal shrinking.

Since \(\( b = \frac{8}{17} \approx 0.47 < 1 \)\), it represents a horizontal stretching of the graph.

### STEP 3: Find the Amplitude of \(\( y \)\)
The amplitude is the absolute value of the coefficient \(\( a \)\) (which is always positive because amplitude measures the distance from the maximum to the baseline).

For our function:
\(\[ a = -1 \]\)
Thus, the amplitude is:
\(\[ \text{Amplitude} = |a| = 1 \]\)

#### Amplitude Sign Description
The amplitude will always be positive because amplitude measures a distance.

### Summary
- Period: \(\( 13.351768777756622 \)\)
- Horizontal Change: Horizontal stretching
- Amplitude: \(\( 1 \)\)
- Amplitude Sign Description: Positive, because amplitude measures a distance.

I hope this helps clarify the steps needed to analyze the trigonometric function \(\( y = -\cos\left(\frac{8x}{17}\right) \)\)!

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