Y=(x+3)^-2

Here are the key properties and quick derivatives for y = (x+3)^-2 = 1/(x+3)^2. - Domain: x ≠ -3 - Vertical asymptote: x = -3 - Horizontal asymptote: y = 0 (as x → ±∞) - Range: y > 0 (since a square) - First derivative (slope): y' = -2/(x+3)^3 - Monotonicity: increasing on (-∞, -3); decreasing on (-3, ∞) - No critical points (y' ≠ 0 for any x ≠ -3) - Second derivative (concavity): y'' = 6/(x+3)^4 > 0 for all x ≠ -3 - Concave up on both intervals (-∞, -3) and (-3, ∞) - Intercepts: - y-intercept: at x = 0, y = 1/9, so (0, 1/9) - x-intercept: none (y never equals 0) - Example tangent line (at x = 0): - f(0) = 1/9, f'(0) = -2/27 - Tangent: y = 1/9 - (2/27)x - Antiderivative (optional): ∫(x+3)^-2 dx = -1/(x+3) + C If you want, I can plot the graph, or work through a specific problem (e.g., tangent at another x, optimization, or area under a curve). What would you like to do?