Problem 56 - Algebra and Trigonometry

Question

Use a vertical shift to graph one period of the function. $$y=\cos x+3$$

Step-by-Step Solution

Answer

The graph of \(y=\cos x+3\) is a cosine graph shifted vertically upward by 3 units. The maximum and minimum of the function shifted to 4 and 2 respectively.

Step 1: Identify the original function and shift

The function \(y=\cos x+3\) is a cosine function that has been shifted upwards by 3 units. The '3' does not change the shape or the period of the function, only its position on the graph. This function will still oscillate; however, its maximum and minimum values will now be 4 and 2, instead of 1 and -1. In other words, the function has been lifted by three units.

Step 2: Plot the basic cosine function

Now, draw the basic cosine curve. The cosine function begins at a maximum value at \(x=0\), descends to its minimum at \(\pi\) (180 degrees), and returns to its maximum at \(2\pi\) (360 degrees). The function repeats this cycle over and over.

Step 3: Add the vertical shift

Next, the entire cosine curve needs to be shifted upwards by three units. This means that the maximum points on the graph at \(y=1\) will now be at \(y=4\), and the minimums at \(y=-1\) will now be at \(y=2\). All points on the original cosine curve are simply lifted by three units.

Step 4: Draw the shifted graph

Finally, draw the grid lines and the cosine curve on the same plot. Remember to label all key points (maximums, minimums, zeros) with their exact coordinates.