Question

Graphing sine and cosine functions Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. $$p(x)=3 \sin (2 x-\pi / 3)+1$$

Short answer

Question: Graph the function $$p(x) = 3\sin(2x-\frac{\pi}{3})+1$$ and determine its amplitude, period, phase shift, and vertical shift. Answer: The graph of the given function has an amplitude of 3, a period of π, a phase shift of \(\frac{\pi}{6}\) units to the right, and a vertical shift of 1 unit upward.

Step-by-step solution

Identify the constants A, B, C, and D

The given function is $$p(x) = 3\sin(2x-\frac{\pi}{3})+1$$Comparing this function to the general form $$p(x) = A \sin(Bx+C) + D$$, we can identify the constants as follows: - \(A=3\): the amplitude - \(B=2\): related to the period - \(C=-\frac{\pi}{3}\): related to the phase shift - \(D=1\): the vertical shift

Calculate amplitude, period, phase shift, and vertical shift

Using the constants, we calculate the following properties of the sine function: - Amplitude (\(A\)): \(|3|=3\) - Period (\(T\)): \(\frac{2\pi}{|2|} = \pi\) - Phase Shift (\(\phi\)): Solve the equation \(2x - \frac{\pi}{3} = 0\), such that \(x=\frac{\pi}{6}\). Therefore, the graph shifts \(\frac{\pi}{6}\) units to the right. - Vertical Shift (\(D\)): \(1\) unit upward

Graph the given function

Using the calculated properties, we can now sketch the sine function. To do so, follow these steps: 1. Plot the basic sine function curve starting from the phase shift value (\(\frac{\pi}{6}\)). 2. Adjust the amplitude by scaling the function vertically by a factor of \(3\). 3. Shift the curve one unit upward to account for the vertical shift. 4. Finally, adjust the period by compressing or stretching the graph horizontally. In this case, the period is shortened to \(\pi\).

Use a graphing utility to check your work

Use a graphing calculator or online tool, such as Desmos or WolframAlpha, to graph the given function $$p(x) = 3\sin(2x-\frac{\pi}{3})+1$$ Check that the graph generated matches the graph obtained in Step 3, taking into consideration amplitude, period, phase shift, and vertical shift.

Key concepts

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function, such as sine or cosine, represents the height difference between the peak and the midpoint of the wave. In a graph of the function \(y = A \times \text{trig}(x)\), where 'trig' stands for either sine or cosine and \(A\) is a constant, the amplitude is the absolute value of \(A\). So, for \(y = 3 \times \text{sin}(x)\), the amplitude would be \(|3| = 3\). This number indicates how 'tall' and 'deep' the waves of the sine or cosine function are.

When graphing, you'll notice that an increase in the amplitude results in a taller wave, and a decrease results in a shorter wave. This scaling transformation doesn't affect the horizontal position of the wave, but it does stretch or compress it vertically. Always remember: the amplitude is positive, even if the value of \(A\) is negative, because it represents a distance which cannot be negative.

Period of Sine and Cosine

The period of sine and cosine functions is the distance along the x-axis required for the function to complete one full cycle of its repeating pattern. For the basic functions \(y = \text{sin}(x)\) and \(y = \text{cos}(x)\), the period is \(2\text{pi}\). However, when these functions are multiplied by a constant \(B\), as in \(y = \text{sin}(Bx)\) or \(y = \text{cos}(Bx)\), the period changes according to the formula \(T = \frac{2\text{pi}}{|B|}\).

For instance, in the function \(y = 3\text{sin}(2x)\), the constant \(B\) is 2, so the period becomes \(\frac{2\text{pi}}{2} = \text{pi}\), which means the sine wave will complete its cycle in half the distance it normally would. This effect either compresses or stretches the graph horizontally. It's vital to understand that changing the period does not move the graph left or right; it simply alters the width of one complete wave cycle.

Phase Shift of Trigonometric Functions

Phase shift in trigonometric functions is when the function's graph is moved horizontally along the x-axis. If our sine or cosine function looks like \(y = \text{trig}(Bx - C)\), the phase shift will be calculated by solving \(Bx - C = 0\) for \(x\). The resulting value of \(x\) is the amount of units the graph will shift. If \(x\) is positive, the graph shifts to the right, and if it's negative, to the left.

Take the function \(y = 3\text{sin}(2x - \frac{\text{pi}}{3})\) as an example. By setting up the equation \(2x - \frac{\text{pi}}{3} = 0\) and solving for \(x\), we find \(x = \frac{\text{pi}}{6}\). This means the graph of the sine function shifts \(\frac{\text{pi}}{6}\) units to the right. Recognizing the phase shift is crucial for plotting the starting point of the cycle of the sine or cosine wave.

Vertical Shift of Trigonometric Functions

A vertical shift in a trigonometric function is a transformation that moves the graph up or down along the y-axis. In the function \(y = A\times \text{trig}(Bx - C) + D\), the value of \(D\) tells us the vertical shift. A positive \(D\) raises the graph above the x-axis, while a negative \(D\) lowers it below the x-axis.

Looking at \(y = 3\text{sin}(2x - \frac{\text{pi}}{3}) + 1\), the vertical shift is \(+1\). This means that every point on the sine graph will be lifted one unit higher than it would be in the absence of the shift. Remember, when graphing, the vertical shift doesn't change the shape or the horizontal position of the graph; it simply elevates or depresses the whole wave uniformly, which can be the difference between a correct solution and a miscalculated one.