Questions: Three graphs are given below. For each, choose its equation from the following. y=2 sec x y=sec (x+π/4) y=tan x y=cot (x-π/4) y=-2+csc x y=-cot x Equation: (Choose one) Equation: (Choose one) Equation: (Choose one)

Solution Steps

The first graph resembles a secant or cosecant function. The graph has asymptotes at $x = \frac{\pi}{4}$ and $x = \frac{3\pi}{4}$, which is a period of $\pi/2$. The standard period of secant and cosecant is $2\pi$. Since $\frac{2\pi}{B} = \frac{\pi}{2}$, we have $B=4$. The graph has a minimum at $y=2$ and a maximum at $y=-2$, so the amplitude is 2. The graph appears to be $2 \sec 4x$.

The second graph resembles a secant or cosecant graph shifted vertically. The graph appears to be shifted down by 2 units. The asymptotes are at $0$ and $\pi$, so the period is $\pi$. Thus $B=2$. The graph resembles $-2 + \csc 2x$.

The third graph resembles a tangent or cotangent graph. The graph appears to be of cotangent since it is decreasing. The asymptotes are at $0$ and $\frac{\pi}{2}$, meaning the period is $\frac{\pi}{2}$. Thus $B=2$. The graph resembles $\cot 2x$. However, $\cot x$ goes to positive infinity as $x$ approaches 0 from the right, and the given graph goes to negative infinity as $x$ approaches 0 from the right. Thus, the graph is $-\cot 2x$.

Final Answer: