To determine which equation matches the graph, let's analyze the graph and each option.

Graph Observations:

• The graph shows a V-shape, which is characteristic of an absolute value function.
• The vertex of the graph is at the point $(0, 0)$.
• The graph opens downwards, indicating that the absolute value function has a negative slope.
• The slope of the arms seems to be $-\frac{1}{3}$ (for both the left and right sides of the vertex).

Now, let’s break down each equation:

Equation A:

$y = -\left| \frac{1}{3} x \right| + 2$

• The absolute value function opens downwards due to the negative sign.
• The graph is shifted upward by 2 units, as indicated by the "+2".
• This does not match the graph, since the vertex is at $(0,0)$.

Equation B:

$y = \left| -\frac{1}{3}x \right| - 2$

• The absolute value function opens upwards because there's no negative outside the absolute value.
• The graph is shifted down by 2 units, as indicated by the "-2".
• This does not match the graph because the graph opens downward and is not shifted down.

Equation C:

$y = -\left| \frac{1}{3} x - 2 \right|$

• The absolute value function opens downwards due to the negative sign.
• The expression $\left| \frac{1}{3} x - 2 \right|$ shifts the vertex horizontally, which we do not observe in the graph.
• This does not match the graph, as the vertex is at the origin.

Equation D:

$y = -\frac{1}{3} \left| x + 2 \right|$

• The absolute value function opens downwards.
• The $\left| x + 2 \right|$ shifts the vertex horizontally to the left by 2 units, which is incorrect for the graph we observe.
• This does not match the graph.

Conclusion:

Based on this analysis, none of the equations exactly match the given graph. However, the correct equation should be something like $y = -\frac{1}{3}|x|$, but with no vertical or horizontal shift (which isn't given in the options).

Let me know if you'd like further details!

Here are 5 related questions to think about:

1. How does the absolute value affect the graph of a function?
2. What role does the negative sign in front of the absolute value play?
3. How can we determine the slope from an absolute value graph?
4. What transformations occur when we add constants outside or inside the absolute value?
5. How would a shift in the vertex change the equation of an absolute value function?

Tip: Always check the vertex position and slope direction to identify transformations in absolute value graphs.