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Chapter 5: Problem 56

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \cos \left(\sin ^{-1} 2 x\right) $$

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In Exercises \(113-116\), use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. In Exercises \(113-114\), convert each angle to a decimal in degrees. Round your answer to two decimal places. $$ 65^{\circ} 45^{\prime} 20^{\prime \prime} $$

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