Answer:
a. [tex]x^2-4x-5= \left(x-2\right)^2-9[/tex].
b. The axis of symmetry for [tex]{\left(x - 2\right)^{2}} - 9[/tex] is [tex]x=2[/tex].
Step-by-step explanation:
a. The vertex form of a quadratic is given by [tex]y=a(x-h)^2+k[/tex], where (h, k) is the vertex.
To convert from [tex]y=x^2-4x-5[/tex] form to vertex form you use the process of completing the square.
Step 1: Write [tex]x^2-4x-5[/tex] in the form [tex]x^2+2ax+a^2[/tex]. Add and subtract 4:
[tex]x^{2} - 4 x - 5=x^{2} - 4 x + {\left(4\right)} - {4} - 5[/tex]
Step 2: Complete the square [tex]x^2+2ax+a^2=\left(x+a\right)^2[/tex]
[tex]{\left(x^{2} - 4 x + 4\right)} - 9={\left(x - 2\right)^{2}} - 9[/tex]
b. The graph of a quadratic function is a parabola. The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. The axis of symmetry always passes through the vertex of the parabola. The x-coordinate of the vertex is the equation of the axis of symmetry of the parabola.
For a quadratic function in standard form, [tex]y=a(x-h)^2+k[/tex], the axis of symmetry is [tex]x=h[/tex].
The axis of symmetry for [tex]{\left(x - 2\right)^{2}} - 9[/tex] is [tex]x=2[/tex].
Look at the graph shown below.
