Question

The function f (x) = x2 + 4x -5 represents a parabola. Part A: Rewrite the function by completing the square. Show ALL your work. Part B: Determine the vertex and explain how you know if it is a maximum or minimum value. Part C: Determine the axis of symmetry for

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f(x).

a) [Answers for Part A, B, and C]
b) [Answers for Part A, B, and C]
c) [Answers for Part A, B, and C]
d) [Answers for Part A, B, and C]

Answer 1

The function f(x) = x^2 + 4x - 5 is rewritten as f(x) = (x + 2)^2 - 9 by completing the square. The vertex is at (-2, -9), and it is a minimum because the parabola opens upwards. The axis of symmetry is x = -2.

Step-by-step explanation:

To rewrite the function f(x) = x^2 + 4x - 5 by completing the square, we want it in the form f(x) = a(x-h)^2 + k, where (h,k) is the vertex of the parabola:

1. Group the x-terms: x^2 + 4x ...
2. Find the value that completes the square: (4/2)^2 = 4.
3. Add and subtract this value inside the parenthesis: x^2 + 4x + 4 - 4.
4. Rewrite this as a perfect square: (x + 2)^2 and carry over the remaining terms: (x + 2)^2 - 4 - 5.

The function becomes f(x) = (x + 2)^2 - 9.

For Part B, the vertex is at (h,k) = (-2, -9). Because a = 1 is positive in (x + 2)^2, the parabola opens upwards, making -9 the minimum value.

For Part C, the axis of symmetry is x = h, thus x = -2.