Given the parabola $f\left(x\right) = x^{2} − 2x − 3$, complete parts (a) through (c) below.

1. Find the vertex by averaging the x-intercepts.

   Hint (a):

   Find the x-intercepts by factoring and using the Zero Product Property. Average the two numbers you get. That x-value represents the line of symmetry and the x-coordinate of the vertex.

   More Help (a):

   Substitute the x-value you got into the equation to find the y-coordinate of the vertex.

2. Find the vertex by completing the square.

   Hint (b):

   What do you need to add to x² − 2x to make it a perfect square? Remember to add it to both sides of the equation and subtract it from the equation so you keep the equation balanced.

   More Help (b):

   When you rewrite the perfect square in factored form, your equation will be in vertex form: $y = a\left(x − h\right)^{2} + k$, where the vertex is (h, k).

3. Find the vertex of $f\left(x\right) = x^{2} + 5x + 2$ using your method of choice.

   Answer (c):

   $\left(−2.5, −4.25\right)$

4. What are the domain and range for $f\left(x\right) = x^{2} + 5x + 2$?

   Answer (d):

   Domain: All real numbers
   Range:$y ≥ −4.25$