The graph of \$y = (x - 3)^2 - 1\$ is a parabola with:
- Vertex at \$(3, -1)\$.
- Axis of symmetry \$x = 3\$.
- \$y\$-intercept at \$(0, 8)\$.
- \$x\$-intercepts at \$(2, 0)\$ and \$(4, 0)\$.
The vertex of a parabola in the form \$y = a(x - h)^2 + k\$ is \$(h, k)\$.
The equation \$y = x^2 + bx + 7\$ can be rewritten in vertex form as \$y = (x + 4)^2 - 9\$.
The vertex form of a quadratic equation is \$y = a(x - h)^2 + k\$, where \$(h, k)\$ is the vertex.
The equation \$y = x^2 - 10x + 23\$ can be converted to vertex form by completing the square:
- \$y = (x - 5)^2 - 2\$.
Completing the square involves adding and subtracting the same value inside the parentheses to create a perfect square trinomial.
The focus is at \$(4, 3)\$ and the directrix is \$y = 1\$.
The vertex is halfway between the focus and directrix, at \$(4, 2)\$.
The equation of the parabola is \$y = \frac{1}{4}(x - 4)^2 + 2\$.
The equation of a parabola with vertex \$(h, k)\$ and focus \$(h, k + p)\$ is \$y = \frac{1}{4p}(x - h)^2 + k\$.
The length of \$AB\$ is \$2\$ because any point on the parabola is equidistant from the focus and the directrix.
The distance from a point on the parabola to the focus is equal to the distance from the point to the directrix.
