1. **Problem statement:** Sketch the graph of the function $$y = 2\sin x - \tan x$$. 2. **Formula and important rules:** - The sine function, $$\sin x$$, oscillates between -1 and 1 with period $$2\pi$$. - The tangent function, $$\tan x$$, has vertical asymptotes where $$\cos x = 0$$, i.e., at $$x = \frac{\pi}{2} + k\pi$$ for integers $$k$$. - The function $$y = 2\sin x - \tan x$$ combines these behaviors. 3. **Key points and behavior:** - At $$x=0$$, $$y = 2\sin 0 - \tan 0 = 0 - 0 = 0$$. - Near vertical asymptotes $$x = \pm \frac{\pi}{2}$$, $$\tan x$$ approaches $$\pm \infty$$, so the function will have vertical asymptotes there. 4. **Sketching steps:** - Plot the sine wave scaled by 2: $$2\sin x$$. - Identify vertical asymptotes at $$x = \frac{\pi}{2} + k\pi$$. - For values between asymptotes, calculate $$y$$ at sample points to understand the curve shape. 5. **Summary:** The graph oscillates with sine shape scaled by 2 but is distorted by the subtraction of $$\tan x$$, which causes vertical asymptotes and sharp changes near $$x = \frac{\pi}{2} + k\pi$$. Final answer: The graph of $$y = 2\sin x - \tan x$$ has vertical asymptotes at $$x = \frac{\pi}{2} + k\pi$$ and oscillates with sine wave shape modified by the tangent subtraction.