Problem 46 - Algebra 2

Question

a. Write \(y=3 \sin (2 x-4)+1\) in the form \(y=a \sin b(x-h)+k\) (Hint Factor where possible.) b. Find the amplitude and period. Describe any translations.

Step-by-Step Solution

Answer

The amplitude is 3. The period is \(\pi\). The graph of the function is shifted 2 units to the right and 1 unit upward.

Step 1: Transformation

As given, the sin function is \( y = 3sin(2x - 4) + 1 \). To get it in the form of \( y = a sin b(x - h) + k \), transform the inner term: \( y = 3 sin(2(x - 2)) + 1 \).

Step 2: Identify the parameters a,b,h and k

From the transformed function \( y = 3sin(2(x - 2)) + 1 \), one can take \( a = 3, b = 2, h = 2, k = 1 \). This format describes all the transformations done on the base sine function.

Step 3: Determine Amplitude, Period and translations

Amplitude (a) is 3. Period (p) is given by \( \frac{2\pi}{b} = \frac{2\pi}{2} = \pi \). It means the function completes its cycle in an interval of \(\pi\). The graph of the function is shifted 2 units to the right (horizontal translation=h), and 1 unit upward (vertical translation=k). Therefore, the sin function is horizontally shifted by 2 units to the right, vertically shifted by 1 unit upward, and has an amplitude of 3 and period of \(\pi\).

Key Concepts

Amplitude

When dealing with trigonometric functions like sine and cosine, the amplitude is a crucial factor. It is the measure of how "tall" the wave gets. In simple terms, the amplitude of a function describes the maximum distance the wave reaches from its equilibrium (center) position. For the function given, \( y=3 \sin(2x - 4) + 1 \), the amplitude is represented by the coefficient in front of the sine function. Here, the coefficient is 3, indicating an amplitude of 3.

• Amplitude determines the height of the wave from the middle to the peak or trough.
• It only affects the vertical stretching and compressing of the wave, not its horizontal aspect.
• A larger amplitude means a taller wave.

In essence, if we visualize the sine wave, each peak will be 3 units away from its center line both upwards and downwards.

Period

The period of a trigonometric function is the distance over which the wave pattern repeats itself. For a sine or cosine function in the form \( y = a \sin b(x - h) + k \), the period can be calculated using the formula \( \frac{2\pi}{b} \). In the function \( y = 3 \sin(2x - 4) + 1 \), the value of \( b \) is 2. This means the period is \( \frac{2\pi}{2} = \pi \).

• The period of the wave gives us the length required for one complete cycle of the wave to be completed.
• For our function, the sine wave will repeat every \( \pi \) units along the x-axis.
• A shorter period indicates a more frequent repetition, thus more oscillations in the same length of x-axis.

Visualizing this, the wave returns to its starting point after every \( \pi \) along the x-axis, completing a full oscillation.

Phase Shift

Phase shift refers to the horizontal movement of the wave along the x-axis. It determines how far a trigonometric function like sine or cosine is shifted from its usual start point. In the equation \( y = 3 \sin(2(x - 2)) + 1 \), the term \( (x - h) \) where \( h = 2 \), represents this horizontal shift.

• Phase shift indicates how much the wave is moved left or right from the origin.
• If \( h \) is positive, the wave shifts to the right, and if negative, to the left.
• For our function, the sine wave is shifted 2 units to the right.

This means if you start from \( x = 0 \), you will find the beginning of our shifted wave at \( x = 2 \). This alteration does not affect the shape of the wave, just its starting position.

Vertical Shift

Vertical shift describes how much a function is moved up or down along the y-axis. This transposition is governed by the constant \( k \) in the function \( y = a \sin b(x - h) + k \). For the function \( y = 3 \sin(2x - 4) + 1 \), \( k \) is 1, signifying a lift of the entire wave by 1 unit upwards vertically.

• Vertical shift moves the sine wave up or down but does not change its shape or orientation.
• A positive \( k \) value indicates an upward shift, while a negative \( k \) suggests a shift downwards.
• For our specific equation, every point on the wave is 1 unit higher than it would typically be without this shift.

This means the baseline of our sine wave is effectively raised, changing the y-values of all points on the function by the same amount.