Question

For each of the following functions, determine the Vertex, Axis of symmetry, Minimum or maximum value, Domain, and Range.

a. f(x)=2(x−4)² +3
b. f(x) = -(x + 3)² - 2.5
c. f(x) = -2(x - 6)²

Answer 1

For the quadratic functions provided, the vertices, axes of symmetry, minimum/maximum values, domains, and ranges were determined.

Step-by-step explanation:

For each of the given quadratic functions, we'll determine the Vertex, Axis of symmetry, Minimum or Maximum value, Domain, and Range:

1. f(x)=2(x−4)² +3

   • The vertex, found by completing the square, is at (4, 3).
   • Axis of symmetry: x=4.
   • Since the coefficient of (x-4)2 is positive, this parabola has a minimum value at the vertex, which is 3.
   • Domain: all real numbers (-∞, +∞).
   • Range: [3, +∞) since it opens upwards.

2. f(x) = -(x + 3)² - 2.5

   • Vertex: (-3, -2.5).
   • Axis of symmetry: x=-3.
   • Maximum value is -2.5 at the vertex because the parabola opens downwards.
   • Domain: all real numbers (-∞, +∞).
   • Range: (-∞, -2.5] since it opens downwards.

3. f(x) = -2(x - 6)²

   • Vertex: (6, 0).
   • Axis of symmetry: x=6.
   • This function has a maximum value of 0 at the vertex.
   • Domain: all real numbers (-∞, +∞).
   • Range: (-∞, 0] because it opens downwards.