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Choose the function(s) that would have the following end behavior. Select all that apply. as x→∞, f(x) → −∞...

Choose the function(s) that would have the following end behavior. Select all that apply.
as x→∞, f(x) → −∞
as x→−∞, f(x) → −∞
□ y = -\frac{2}{3}(x + 7)^2 - 9
□ y = -x^2 + 6x - 8
□ y = \frac{1}{2}x^2 - 6x + 3
□ y = (x + 3)(x - 6)
□ y = -(x + 3)^2 - 6

Answer

Answer:

  1. y = -\frac{2}{3}(x + 7)^2 - 9: This function has the form of a downward-opening parabola (negative coefficient for the squared term), so as x→∞, f(x) → −∞ and as x→−∞, f(x) → −∞. Selected.

  2. y = -x^2 + 6x - 8: This is a quadratic function with a negative leading coefficient, indicating a downward-opening parabola. As x→∞, f(x) → −∞ and as x→−∞, f(x) → −∞. Selected.

  3. y = \frac{1}{2}x^2 - 6x + 3: This is a quadratic function with a positive leading coefficient, indicating an upward-opening parabola. As x→∞, f(x) → ∞ and as x→−∞, f(x) → ∞. Not selected.

  4. y = (x + 3)(x - 6): This is a quadratic function with a positive leading coefficient (when expanded), indicating an upward-opening parabola. As x→∞, f(x) → ∞ and as x→−∞, f(x) → ∞. Not selected.

  5. y = -(x + 3)^2 - 6: This function has the form of a downward-opening parabola (negative coefficient for the squared term), so as x→∞, f(x) → −∞ and as x→−∞, f(x) → −∞. Selected.

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