Question

describing a quadratic function
use the drop - down menus to describe the function $g(x)=x^{2}-14x - 1$.
the vertex of $g$ is
the axis of symmetry is
to graph the function $g$, shift the graph of $f = x^{2}$

Explanation:

Step1: Find vertex x-coordinate

For quadratic $ax^2+bx+c$, $x=-\frac{b}{2a}$. Here $a=1, b=-14$:
$x = -\frac{-14}{2(1)} = 7$

Step2: Find vertex y-coordinate

Substitute $x=7$ into $g(x)$:
$g(7) = 7^2 -14(7) -1 = 49 - 98 -1 = -50$

Step3: Identify axis of symmetry

Axis of symmetry is $x$ of vertex: $x=7$

Step4: Rewrite g(x) for shifts

Complete the square:
$g(x) = x^2-14x -1 = (x^2-14x+49) -49 -1 = (x-7)^2 -50$
Compare to $f(x)=x^2$: right 7, down 50.

Answer:

The vertex of $g$ is $\boldsymbol{(7, -50)}$
The axis of symmetry is $\boldsymbol{x=7}$
To graph the function $g$, shift the graph of $f = x^2$ 7 units to the right and 50 units down