Question

1. which function below has this graph? graph a. $y = \sin x$ b. $y = \tan x$ c. $y = \cos x$ d. none of the above 21. which function below has this graph? graph a. $y = \sec x$ b. $y = \csc x$ c. $y = \cot x$ d. none of the above

Explanation:

Question 20

Step1: Recall Graphs of Trig Functions

• \( y = \sin x \): Oscillates between -1 and 1, has a smooth wave shape (sine curve), no vertical asymptotes.
• \( y = \tan x \): \( \tan x=\frac{\sin x}{\cos x} \), has vertical asymptotes at \( x = \frac{\pi}{2}+k\pi \) (\( k\in\mathbb{Z} \)), and its graph consists of periodic branches with a "S" - like shape in each period, passing through the origin.
• \( y = \cos x \): Oscillates between -1 and 1, smooth wave shape (cosine curve), no vertical asymptotes.

Step2: Analyze the Given Graph

The given graph has vertical asymptotes and the shape of the branches matches the graph of \( y = \tan x \). \( y=\sin x \) and \( y = \cos x \) do not have vertical asymptotes, so they can be eliminated.

Step1: Recall Graphs of Trig Functions

• \( y=\sec x=\frac{1}{\cos x} \): Has vertical asymptotes at \( x=\frac{\pi}{2}+k\pi \) (\( k\in\mathbb{Z} \)), and its graph consists of periodic "U" - shaped and inverted "U" - shaped branches (since \( \sec x\geq1 \) or \( \sec x\leq - 1 \)).
• \( y = \csc x=\frac{1}{\sin x} \): Has vertical asymptotes at \( x = k\pi \) (\( k\in\mathbb{Z} \)), and its graph consists of periodic "U" - shaped and inverted "U" - shaped branches (since \( \csc x\geq1 \) or \( \csc x\leq - 1 \)), but the position of asymptotes is different from \( \sec x \).
• \( y=\cot x=\frac{\cos x}{\sin x} \): Has vertical asymptotes at \( x = k\pi \) (\( k\in\mathbb{Z} \)), and its graph has a different shape (more like a "hyperbola - like" shape in each period) compared to the given graph.

Step2: Analyze the Given Graph

The given graph has vertical asymptotes at \( x=\frac{\pi}{2}+k\pi \) (consistent with \( \sec x \)'s asymptotes) and the "U" - shaped and inverted "U" - shaped branches match the graph of \( y=\sec x \). The graph of \( y = \csc x \) has asymptotes at \( x = k\pi \), and \( y=\cot x \) has a different shape, so they can be eliminated.

Answer:

B. \( y = \tan x \)

Question 21