Tangent Line 06E365

1. **State the problem:** Find the equation of the tangent line to the function $y = \tan x$ at $x = -\frac{\pi}{4}$. Then graph both the function and the tangent line. 2. **Recall the formula for the tangent line:** The tangent line to $y = f(x)$ at $x = a$ is given by: $$y = f(a) + f'(a)(x - a)$$ where $f'(a)$ is the derivative of $f$ evaluated at $x = a$. 3. **Find $f(a)$:** $$f\left(-\frac{\pi}{4}\right) = \tan\left(-\frac{\pi}{4}\right) = -1$$ 4. **Find the derivative $f'(x)$:** $$f'(x) = \sec^2 x$$ 5. **Evaluate the derivative at $x = -\frac{\pi}{4}$:** $$f'\left(-\frac{\pi}{4}\right) = \sec^2\left(-\frac{\pi}{4}\right) = \left(\frac{1}{\cos\left(-\frac{\pi}{4}\right)}\right)^2 = \left(\frac{1}{\frac{\sqrt{2}}{2}}\right)^2 = (\sqrt{2})^2 = 2$$ 6. **Write the equation of the tangent line:** $$y = f(a) + f'(a)(x - a) = -1 + 2\left(x + \frac{\pi}{4}\right)$$ 7. **Simplify the tangent line equation:** $$y = -1 + 2x + \frac{\pi}{2} = 2x - 1 + \frac{\pi}{2}$$ **Final answer:** $$\boxed{y = 2x - 1 + \frac{\pi}{2}}$$ 8. **Graphing:** The function $y = \tan x$ and the tangent line $y = 2x - 1 + \frac{\pi}{2}$ should be graphed on the same axes with window $[-2\pi, 2\pi]$ for $x$ and $[-4,4]$ for $y$.