How do you graph #y=secx+2#?

Question

Victoria Adebayo · Accepted Answer

See answer below

Given: #y = sec x + 2#

First draw a dashed vertical shift line at #y = 2#

Since #sec x = 1/(cos x)#, sketch a dashed cosine function

#y = cos x + 2 => " amplitude" = 1 " and period " = 2 pi#

Remember that a cosine with a period of #2 pi# needs to be divided into 4 sections: #0, pi/2, pi, (3pi)/2, 2 pi#.

Wherever the cosine function crosses the #y = 2# line there will be a vertical asymptote. At each peak and trough, there will be a point on the secant function that arcs up to the adjacent vertical asymptotes:

Grace Ashcraft · Answer

undefined To graph $ y = \sec(x) + 2 $, follow these steps:

1. Identify the vertical asymptotes: Vertical asymptotes occur where the secant function has undefined values, which are at odd multiples of $ \pi/2 $. So, the vertical asymptotes are at $ x = \frac{\pi}{2} + n\pi $ and $ x = -\frac{\pi}{2} + n\pi $, where $ n $ is an integer.

2. Determine the behavior near the vertical asymptotes: As $ x $ approaches the vertical asymptotes, the function approaches positive or negative infinity, depending on the side of the asymptote.

3. Find the x-intercepts: The x-intercepts occur where $ \sec(x) + 2 = 0 $. Since $ \sec(x) = \frac{1}{\cos(x)} $, this equation has no real solutions because $ \sec(x) $ is always positive or zero.

4. Determine the behavior as $ x $ approaches positive or negative infinity: As $ x $ approaches positive or negative infinity, $ \sec(x) $ oscillates between positive and negative infinity, so $ \sec(x) + 2 $ also approaches positive and negative infinity.

5. Plot some key points: Choose some values of $ x $ to evaluate $ y = \sec(x) + 2 $, such as $ x = 0, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, $ and $ x = \pi $. Then, plot the corresponding points on the graph.

6. Draw the graph: Connect the points smoothly, taking into account the behavior near vertical asymptotes and at infinity.

7. Label the graph: Label the vertical asymptotes and any other key features, such as points of inflection or regions of increasing/decreasing behavior.

Following these steps will help you accurately graph the function $ y = \sec(x) + 2 $.