Question

For the function y=x^2-4x-5, perform the followingtasks:

Answer 1

Answer:

a. $x^2-4x-5= \left(x-2\right)^2-9$.

b. The axis of symmetry for ${\left(x - 2\right)^{2}} - 9$ is $x=2$.

Step-by-step explanation:

a. The vertex form of a quadratic is given by $y=a(x-h)^2+k$, where (h, k) is the vertex.  

To convert from $y=x^2-4x-5$ form to vertex form you use the process of completing the square.

Step 1: Write $x^2-4x-5$ in the form $x^2+2ax+a^2$. Add and subtract 4:

$x^{2} - 4 x - 5=x^{2} - 4 x + {\left(4\right)} - {4} - 5$

Step 2: Complete the square $x^2+2ax+a^2=\left(x+a\right)^2$

${\left(x^{2} - 4 x + 4\right)} - 9={\left(x - 2\right)^{2}} - 9$

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, $y=a(x-h)^2+k$, the axis of symmetry is $x=h$.

The axis of symmetry for ${\left(x - 2\right)^{2}} - 9$ is $x=2$.

Look at the graph shown below.