Analyzing Graph Coordinates: Understanding (3,3) and (9,3)

The graph corresponds to

The problem involves determining the equations corresponding to a blue absolute value graph and a red horizontal line based on a given graph.

Firstly, consider the blue absolute value graph:

• It is symmetric and has a "V" shape, typically expressed as $f(x) = |x - h| + k$.
• The vertex, or minimum point, of this graph is at (6, 6). Hence, the equation can be written as $f(x) = |x - 6|$, because at $x = 6$, the function value is at its minimum.
• The value of $k$, the vertical shift, is not explicitly needed here since the vertex already indicates the graph reaches down to the x-axis and $f(x) = |x - 6| + 0$.

Next, assess the red horizontal line:

• This line crosses the y-axis at $y = 3$, which aligns with the horizontal segment at points (3, 3) and (9, 3).
• Thus, the constant function equation is $g(x) = 3$.

Upon evaluation of the given choices, choice 1 corresponds correctly with these interpretations:

$\begin{cases} f(x)= |x-6| \\ g(x)=3 \end{cases}$

Therefore, the correct set of functions is $f(x) = |x - 6|$ and $g(x) = 3$.