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?