Question

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=\cos x, \quad y=\frac{1}{2} \cos \left(x-\frac{\pi}{4}\right)\)

Short answer

The first function, \(y = \cos x\), has an amplitude of 1, and a period of \(2\pi\). Plot points like (-pi/2, 0), (0,1), (pi/2, 0), (pi, -1), (3pi/2, 0), and (2pi, 1). The second function, \(y = \frac{1}{2} \cos\left(x-\frac{\pi}{4}\right)\), has half the amplitude and the same period as the first function but is shifted right by \(\frac{\pi}{4}\). Apply these transformations to the points of the first function, and plot points like (-pi/4, 0), (pi/4, 1/2), (3pi/4, 0), (5pi/4, -1/2), (7pi/4, 0), and (9pi/4, 1/2) to sketch the graph of the second function.

Step-by-step solution

Analyze the first function \(y = \cos x\)

For the function \(y = \cos x\), we can notice that: 1. Amplitude: 1 (since it’s not multiplied by any constant) 2. Period: \(2\pi\) (since \(B = 1\); period is given by \(\frac{2\pi}{B}\)) 3. Phase shift: none (since there is no constant inside the cosine function) 4. Vertical shift: none (since there is no constant summed outside the cosine function) Let's plot some important points: (-pi/2, 0), (0,1), (pi/2, 0), (pi, -1), (3pi/2, 0) and (2pi, 1)

Analyze the second function \(y = \frac{1}{2} \cos\left(x-\frac{\pi}{4}\right)\)

For the function \(y = \frac{1}{2} \cos\left(x-\frac{\pi}{4}\right)\), we can notice that: 1. Amplitude is multiplied by a factor of 1/2: The graph of the second function will be "shorter" vertically. 2. Period: \(2\pi\) (same period as the first function since it is not multiplied by any constants) 3. Phase shift: \(\frac{\pi}{4}\) to the right (since \(C = -\frac{\pi}{4}\); the values of the function are "shifted" to the right by \(\frac{\pi}{4}\)) 4. Vertical shift: none (same as in the first function, there is no constant summed outside the cosine function)

Use transformation(s) to obtain the graph of the second function from the graph of the first function

From the analysis in step 1 and step 2, we can understand that to sketch the second function, we need to do two transformations to the graph of the first function: 1. Multiply the amplitude of function \(y=\cos x\) by 1/2. 2. Shift the function to the right by \(\frac{\pi}{4}\) units. Apply these transformations to the points of the first function: (-pi/2 + pi/4, 0 * 1/2) = (-pi/4, 0) (0 + pi/4, 1 * 1/2) = (pi/4, 1/2) (pi/2 + pi/4, 0 * 1/2) = (3pi/4, 0) (pi + pi/4, -1 * 1/2) = (5pi/4, -1/2) (3pi/2 + pi/4, 0 * 1/2) = (7pi/4, 0) (2pi + pi/4, 1 * 1/2) = (9pi/4, 1/2) We can use these points to sketch the graph of the second function \(y = \frac{1}{2} \cos\left(x-\frac{\pi}{4}\right)\).