We are given the graph of a parabola $g(x)$ which is a transformation of the parent function $f(x) = x^2$. We need to identify the transformations applied, write the equation for $g(x)$ in function notation and as an equation, and determine the domain and range of $g(x)$.

1. Problem Description

We are given the graph of a parabola $g(x)$ which is a transformation of the parent function $f(x) = x^2$. We need to identify the transformations applied, write the equation for $g(x)$ in function notation and as an equation, and determine the domain and range of $g(x)$.

2. Solution Steps

First, let's identify the transformations. The dashed parabola is the graph of $f(x) = x^2$. The solid parabola $g(x)$ appears to be the result of a horizontal shift. The vertex of the dashed parabola is at $(0, 0)$, and the vertex of the solid parabola is at $(3, -4)$. This means that the parabola has been shifted 3 units to the right and 4 units down.

Vertical Shift: Down 4 units.

Horizontal Shift: Right 3 units.

Stretches/Compressions: No stretch or compression, as the basic shape of the parabola remains the same. When $x=0$ in the parent function $f(x)=x^2$, $f(x) = 0$. When $x=1$, $f(x) = 1$. For $g(x)$, when $x=3$, $g(x) = -4$. When $x=4$, $g(x) = -3$. This confirms that there is no vertical stretch or compression.

Thus, $g(x)$ can be written as a transformation of $f(x) = x^2$ as follows:

$g(x) = f(x - 3) - 4$.

In function notation: $g(x) = (x - 3)^2 - 4$.

The domain of a quadratic function is all real numbers. In interval notation, this is $(-\infty, \infty)$.

The range of $g(x)$ is determined by the vertex. Since the vertex is at $(3, -4)$ and the parabola opens upwards, the range is $[-4, \infty)$.

3. Final Answer

1) Choose the correct transformation (Stretches/Compressions): None

2) Choose the correct transformation (Vertical Shifts): Down 4

3) Choose the correct transformation (Horizontal Shifts): Right 3

4) Write $g(x)$ using function notation: $g(x) = (x-3)^2 - 4$

5) Write $g(x)$ as an equation: $g(x) = x^2 - 6x + 5$

6) Write the domain of $g(x)$ using interval notation: Domain of $g(x)$: $(-\infty, \infty)$

7) Write the range of $g(x)$ using interval notation: Range of $g(x)$: $[-4, \infty)$