Question

Victor and Stewart determined the phase shift for the function \(f(x)=4 \sin (2 x-6)+12 .\) Victor said that the phase shift was 6 units to the right, while Stewart claimed it was 3 units to the right. a) Which student was correct? Explain your reasoning. b) Graph the function to verify your answer from part a).

Short answer

Stewart is correct; the phase shift is 3 units to the right.

Step-by-step solution

- Identify the standard form of the sine function

The standard form of the sine function is \[ f(x) = A \sin(Bx - C) + D \]where A is the amplitude, B is the frequency, C/B is the phase shift, and D is the vertical shift. The given function is \[ f(x) = 4 \sin(2x - 6) + 12 \].

- Determine the phase shift

To find the phase shift of the function, use the formula \[ \text{Phase shift} = \frac{C}{B} \].From the function, we can see that B = 2 and C = 6. Therefore, plug in the values to get the phase shift:\[ \text{Phase shift} = \frac{6}{2} = 3 \text{ units to the right} \].

- Compare the calculations with student answers

Victor said the phase shift was 6 units to the right. Stewart claimed it was 3 units to the right. Based on our calculation, the phase shift is 3 units to the right. Therefore, Stewart is correct.

- Graph the function

To verify the phase shift, graph the function \[ f(x) = 4 \sin(2x - 6) + 12 \]. Notice that the graph of the sine function is shifted 3 units to the right compared to the standard \[ \sin(2x) \] function. Additionally, plot the function and check if the maximums, minimums, and intercepts align taking into account a shift of 3 units to the right and a vertical shift of 12.

Key concepts

Amplitude

The amplitude of a sine function is the vertical stretch or compression. It shows how high and low the waves go from the centerline (or midline) of the graph. For the sine function in the standard form \( f(x) = A \sin(Bx - C) + D \), the amplitude is represented by A.
In the function \( f(x) = 4 \sin(2x - 6) + 12 \), the amplitude is 4. This means the sine wave will reach 4 units above and 4 units below its midline.
The amplitude doesn't affect the horizontal position (phase shift) or the function's frequency but changes the height of the wave.

Frequency

Frequency determines how many cycles the sine function completes in a given interval. In the standard form \( f(x) = A \sin(Bx - C) + D \), frequency is represented by B. It tells us how many sine periods fit into \( 2\pi \).
For the function \( f(x) = 4 \sin(2x - 6) + 12 \), frequency is 2. This means the sine wave completes 2 full cycles within \( 2\pi \) units of x.
Higher frequency results in more waves within a given range, making the graph 'tighter'. Lower frequency spreads the waves out further apart.

Vertical Shift

Vertical shift moves the sine function up or down along the y-axis. In the standard form \( f(x) = A \sin(Bx - C) + D \), vertical shift is represented by D.
In the function \( f(x) = 4 \sin(2x - 6) + 12 \), the vertical shift is 12. This means the entire sine wave is moved 12 units upwards from the origin (y=0).
The vertical shift affects the position of the midline of the wave, but does not affect the wave's amplitude or period.

Graphing Trigonometric Functions

Graphing trigonometric functions like sine involves understanding their different parameters: amplitude, frequency, phase shift, and vertical shift.
To graph \( f(x) = 4 \sin(2x - 6) + 12 \), follow these steps:

• Identify the amplitude (4), frequency (2), and vertical shift (12).
• Determine the phase shift using the formula \( \text{Phase shift} = \frac{C}{B} = \frac{6}{2} = 3 \) units to the right.
• Start by plotting the vertical shift. Move the middle line of the sine function up to y=12.
• Plot the basic sine wave shape, considering the amplitude. It will reach up to 4 units above and 4 below the midline at y=12.
• Shift the whole sine wave 3 units to the right.

By following these steps, you create an accurate graph. The sine function \(4 \sin(2x - 6) + 12\) moves 3 units right, stretches vertically by 4, and rises 12 units up.