9514 1404 393
Answer:
3 (7e) f(x) = -3(x +2)^2 +1; D: (-∞, ∞); R: (-∞, 1]
4 (8) g(x) = 4(x -3)^2 -5; x = 3; (3, -5); min: -5; D: (-∞, ∞); R: [-5, ∞)
5 (10) A(175) = 31,250 m^2
Step-by-step explanation:
Q3. (7e) f(x) = -3x^2 -12x -11 = -3(x^2 +4x) -11 = -3(x^2 +4x +4) -11 +3(4)
f(x) = -3(x +2)^2 +1
Domain: all real numbers; Range: y ≤ 1
Expanded: -3(x^2 +4x +4) +1 = -3x^2 -12x -11
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Q4. (8a) g(x) = 4(x^2 -6x) +31 = 4(x^2 -6x +9) +31 -4(9)
g(x) = 4(x -3)^2 -5
8b) x = 3 . . . axis of symmetry
8c) (3, -5) . . . coordinates of vertex
8d) minimum: -5 (parabola opens upward from vertex, which is the minimum)
8e) any quadratic has a domain of "all real numbers"
8f) the range is upward from the minimum: y ≥ -5
8g) see attached
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Q5. (10) A(x) = 700x -2x^2 = -2(x^2 -350x +30625) +2(30625)
A(x) = -2(x -175)^2 +31,250
The maximum area that can be enclosed is 31,250 square meters.
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Additional comment
The process of "completing the square" can be done using the following steps. (a) Factor out the leading coefficient from the variable terms; (b) determine the linear term coefficient inside parentheses, and add the square of half of it inside parentheses; (c) add the opposite of the same quantity outside parentheses (make sure the leading coefficient is properly factored in; (d) rewrite the parentheses as a square, and collect terms outside parentheses. In the final form a(x -h)^2 +k, the vertex is (h, k).
