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f(x)=(x+1)^2
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GPT-5 mini
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Here are some basic facts and forms for f(x) = (x+1)^2:
- Expanded form: f(x) = x^2 + 2x + 1.
- Vertex (minimum): (-1, 0). The parabola opens upward (coefficient of x^2 is positive).
- Axis of symmetry: x = -1.
- Domain: all real numbers, (-∞, ∞).
- Range: f(x) ≥ 0 (i.e., [0, ∞)).
- x-intercept: x = -1 (double root). y-intercept: f(0) = 1.
- Derivative: f'(x) = 2(x+1) = 2x + 2.
- Second derivative: f''(x) = 2 (concave up everywhere).
- Indefinite integral: ∫f(x) dx = (x+1)^3/3 + C = x^3/3 + x^2 + x + C.
Would you like a sketch of the graph, evaluations at particular x-values, or something else?
- Expanded form: f(x) = x^2 + 2x + 1.
- Vertex (minimum): (-1, 0). The parabola opens upward (coefficient of x^2 is positive).
- Axis of symmetry: x = -1.
- Domain: all real numbers, (-∞, ∞).
- Range: f(x) ≥ 0 (i.e., [0, ∞)).
- x-intercept: x = -1 (double root). y-intercept: f(0) = 1.
- Derivative: f'(x) = 2(x+1) = 2x + 2.
- Second derivative: f''(x) = 2 (concave up everywhere).
- Indefinite integral: ∫f(x) dx = (x+1)^3/3 + C = x^3/3 + x^2 + x + C.
Would you like a sketch of the graph, evaluations at particular x-values, or something else?
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