Question

the graph of g(x) is the graph of f(x)=x^2 translated 4 units left and 7 units down. write the function of g(x) in vertex form.

Explanation:

Step1: Recall the vertex - form of a quadratic function

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. The original function is $f(x)=x^2$, which has a vertex at $(0,0)$ (since for $y = x^2$, $a = 1$, $h = 0$, and $k = 0$).

Step2: Apply the translation rules

When a graph is translated $4$ units to the left, we replace $x$ with $(x + 4)$ in the function. When it is translated $7$ units down, we subtract $7$ from the function.
For $f(x)=x^2$, after the left - hand translation of $4$ units, the function becomes $y=(x + 4)^2$. After the downward translation of $7$ units, the function $g(x)$ is $g(x)=(x + 4)^2-7$.

Answer:

$g(x)=(x + 4)^2-7$