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

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 ).