In Exercises 53–60, use a vertical shift to graph one period of the function. y = cos x + 3

Verified step by step guidance

1. Identify the base function and its characteristics. Here, the base function is \(y = \cos x\), which has a period of \(2\pi\), an amplitude of 1, and oscillates between -1 and 1.
2. Understand the effect of the vertical shift. The function \(y = \cos x + 3\) shifts the entire graph of \(\cos x\) upward by 3 units.
3. Determine the new range of the function after the vertical shift. Since the original range of \(\cos x\) is \([-1, 1]\), adding 3 shifts this to \([-1 + 3, 1 + 3]\), which is \([2, 4]\).
4. Sketch one period of the function from \(x = 0\) to \(x = 2\pi\). Plot the key points of \(\cos x\) (such as at \(0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\)) and then shift their \(y\)-values up by 3.
5. Label the graph with the new midline \(y = 3\), the maximum value at \(y = 4\), and the minimum value at \(y = 2\), showing the vertical shift clearly.

Key Concepts

Vertical Shift in Trigonometric Functions

A vertical shift moves the entire graph of a function up or down without changing its shape. For y = cos x + 3, the '+3' shifts the cosine curve upward by 3 units, raising the midline from y = 0 to y = 3.

Period of the Cosine Function

The period of the cosine function is the length of one complete cycle, which is 2π for y = cos x. Graphing one period means plotting the function from 0 to 2π (or any interval of length 2π) to show a full wave.

Graphing One Period of a Trigonometric Function

To graph one period, identify key points such as maxima, minima, and zeros within one cycle. For y = cos x + 3, plot points of cos x shifted up by 3 over the interval [0, 2π], showing the wave oscillating between 4 and 2.