Question

Make a complete graph of each function. Find the amplitude, period, and phase shift. $$y=3 \cos \left(x-\frac{\pi}{4}\right)$$

Short answer

Amplitude: 3, Period: \(2\pi\), Phase Shift: \(\frac{\pi}{4}\) to the right.

Step-by-step solution

Determine the Amplitude

The amplitude of a trigonometric function is the coefficient of the cosine function. In this case, the coefficient of \(\cos\) is 3. Therefore, the amplitude is 3, which is the distance from the midline of the graph to the maximum or minimum value of the function.

Determine the Period

The period of a cosine function is determined by the coefficient of \(x\) inside the argument of the cosine function. The standard period of \(\cos(x)\) is \(2\pi\) and since there is no coefficient other than 1 multiplying \(x\), the period remains unaltered. Thus, the period of our function is \(2\pi\).

Determine the Phase Shift

The phase shift is determined by the horizontal shift inside the argument of the cosine function. The general form is \(\cos(x - c)\), where \(c\) would be the phase shift. In our case, the phase shift is \(\frac{\pi}{4}\) to the right.

Graph the Function

To graph \(y = 3 \cos(x - \frac{\pi}{4})\), start by plotting the phase shift along the x-axis. Next, plot points for the maximum amplitude, minimum amplitude, and midline while considering the period. Complete one cycle of the cosine wave, and then repeat this pattern to fill the graph.

Key concepts

Amplitude of Trigonometric Functions

In trigonometry, the amplitude of a function refers to the height of the wave, or how 'tall' the wave is from its average or rest position. It specifically measures the distance from the midline to the maximum (peak) or minimum (trough) value of the wave. For a function like \(y = A\cos(B(x - C))\) or \(y = A\sin(B(x - C))\), \(A\) represents the amplitude, and it indicates the scalene factor of the wave. When the amplitude is positive, it means that the graph stretches vertically; when negative, it reflects the wave across the horizontal axis.

For example, looking at the function \(y=3\cos\left(x-\frac{\pi}{4}\right)\), the amplitude is given by the coefficient of the cosine function, which in this case is 3. This significant value tells us how much the graph of our function will stretch vertically. If you want the graph's peaks and troughs to precisely represent the function's amplitude, ensure you mark the y-axis accordingly with a range of at least '3' units above and below the midline.

Period of Trigonometric Functions

The period of a trigonometric function describes the length of one full cycle of the wave before it starts repeating. In simpler terms, it's the 'distance' along the x-axis that the function needs to traverse to complete one oscillation. For a standard sine or cosine function such as \(\sin(x)\) or \(\cos(x)\), the period is \(2\pi\), meaning that the pattern of the wave repeats every \(2\pi\) units along the x-axis.

To ascertain the period of a function resembling \(A\cos(Bx - C)\) or \(A\sin(Bx - C)\), divide the standard period \(2\pi\) by the absolute value of \(B\). When \(B\) is 1, the period remains \(2\pi\) as seen in our example \(y=3\cos(x-\frac{\pi}{4})\).

Important Tip:

In an exercise, carefully examining the coefficient of \(x\) within the trigonometric function's argument is crucial as it plays a vital role in determining the period. Remember, a larger value of \(B\) signifies a shorter period leading to a more 'frequent' wave, while a smaller value makes for a more 'stretched-out' wave over the x-axis.

Phase Shift in Trigonometry

Phase shift, in trigonometry, refers to the horizontal displacement of a trigonometric function along the x-axis. Simply put, it's how much the function is shifted left or right from its usual position. A positive phase shift indicates a movement to the right, whereas a negative one shows a movement to the left. For a function outlined as \(y = A\cos(B(x - C))\) or \(y = A\sin(B(x - C))\), the value of \(C\) corresponds to the phase shift.

In the function \(y=3\cos\left(x-\frac{\pi}{4}\right)\), the \(\frac{\pi}{4}\) inside the cosine's argument denotes a phase shift to the right by \(\frac{\pi}{4}\) units. Recognizing the phase shift is essential for accurately graphing the function since it determines where the wave starts on the x-axis.

Quick Tip:

To graph a function with a phase shift, begin at the shifted starting point, then follow your findings for amplitude and period to complete the wave pattern. Always remember to communicate the phase shift on the x-axis when you're sketching the graph to represent the function accurately.