Question

Determine the amplitude and the period for the function. Sketch the graph of the function over one period. $$ y=\cos \left(x-\frac{\pi}{4}\right) $$

Short answer

The amplitude of the given function is 1, and the period is \(2\pi\). To sketch the graph over one period, account for the phase shift of \(-\frac{\pi}{4}\) and mark the highest point at \((\frac{\pi}{4}, 1)\), the lowest point at \((\frac{5\pi}{4},-1)\), and x-axis intersections at \((\frac{\pi}{2}, 0)\) and \((\frac{3\pi}{2}, 0)\). Connect the points with a smooth curve to represent the cosine function.

Step-by-step solution

Identifying the Amplitude

To find the amplitude, we need to analyze the cosine function, which has the general format: \(y = A \cos(Bx + C)\), where A is the amplitude, B determines the period, and C determines the phase shift. The given function is \(y = \cos\left(x - \frac{\pi}{4}\right)\). Here, A = 1, B = 1, and C = -\(\frac{\pi}{4}\). The amplitude is the maximum value of the function minus its minimum value divided by 2, which for a cosine function is given by the absolute value of the coefficient of the cosine function. So, in this case, the amplitude is \(|A| = |1| = 1\).

Determining the Period

The period of the cosine function is the length of one full cycle before it starts to repeat itself. The general formula to find the period of a cosine function is \(T = \frac{2\pi}{|B|}\). In our given function, B = 1, so the period would be \(T = \frac{2\pi}{|1|} = 2\pi\).

Sketching the Graph

Since our function has a period of \(2\pi\) and amplitude of 1, we can create a simple sketch of the graph over one period, taking into account the phase shift of \(-\frac{\pi}{4}\). 1. Begin by drawing the x-axis and y-axis on graph paper. 2. Mark the x-axis with increments that represent \(\frac{\pi}{4}\) units. 3. Since the phase shift is \(-\frac{\pi}{4}\), the function will start at \(\frac{\pi}{4}\) on the x-axis instead of at 0. 4. Draw the highest point (amplitude = 1) at \((\frac{\pi}{4}, 1)\) and the lowest point (amplitude = -1) at \((\frac{5\pi}{4},-1)\). 5. Mark the points where the function intersects the x-axis: \((\frac{\pi}{2}, 0)\) and \((\frac{3\pi}{2}, 0)\). 6. Draw a smooth curve connecting the points, making a wave pattern that resembles a cosine function. You should now have a sketch of the given function over one period, with an amplitude of 1 and a period of \(2\pi\).

Key concepts

Amplitude of Cosine Function

When we talk about the \textbf{amplitude} of a cosine function, we're referring to the height of its peaks and the depth of its valleys from the central line, usually the x-axis. In a formula like y = A cos(Bx + C), the amplitude is given by the absolute value of the coefficient A.

In our example, y = cos(x - \frac{\(pi\)}{4}), the amplitude is simply 1 since A=1. It tells us how 'tall' or 'intense' the waves of our function are, with higher values indicating taller waves. The key point to remember is that the amplitude is always a positive value; even if A were negative, we'd take the absolute value to find the amplitude.

Period of Cosine Function

The \textbf{period} of a function is the distance over the x-axis that the function takes to start repeating its pattern. For trigonometric functions like the cosine function, this typically presents as a full wave cycle.

Using the formula, T = \(frac{2\(pi\)}{|B|}\), where B affects the period, we can determine the length of one full cycle. For y = cos(x - \(frac{\(pi\)}{4}\)), since B=1, the period is 2\(pi\), which means that after an x-advance of 2\(pi\), the cosine wave starts to repeat its form.

Phase Shift

The concept of \textbf{phase shift} refers to the horizontal shift of the graph of a trigonometric function along the x-axis. It represents how much the entire wave is moved either to the left or to the right.

In the general formula y = A cos(Bx + C), the phase shift is determined by the value of C. If C is positive, the graph shifts to the left, and if it's negative, it shifts to the right. For our function, y = cos(x - \(frac{\(pi\)}{4}\)), C is -\(frac{\(pi\)}{4}\), which means our graph shifts to the right by \(frac{\(pi\)}{4}\), starting its first peak at \(frac{\(pi\)}{4}\) rather than at 0.

Sketching Cosine Graphs

When \textbf{sketching cosine graphs}, it's crucial to accurately capture the amplitude, period, and phase shift properties on a graph to visualize the function's behavior. Here's how to sketch our example function, y = cos(x - \(frac{\(pi\)}{4}\)), following these steps:

• Draw both x and y axes, labeling x-axis increments in terms of radians or degrees, as appropriate.
• Note the amplitude on the y-axis, marking the highest and lowest points the function will reach.
• Mark one period length on the x-axis based on the period calculation.
• Account for phase shift by starting the initial point of the wave at its respective shift along the x-axis.
• With the amplitude and period as your guide, plot the characteristic 'peak' and 'trough' points of the cosine wave.
• Finally, draw a smooth curve through the points, ensuring the wave aligns with the phase shift and oscillates with the right amplitude.

After these steps, the resulting visual is a clear representation of the cosine function across one full cycle.