Based on the graph, we need to identify which graph corresponds to each of the power functions listed:

1. $f(x) = x$ (linear function)
2. $f(x) = x^3$ (cubic function)
3. $f(x) = x^5$ (quintic function)
4. $f(x) = x^{-2}$ (inverse square function)
5. $f(x) = x^{1/2}$ (square root function)

Analyzing the Graphs:

• Graph A (Black): This graph decreases rapidly as $x$ approaches zero from the right and decreases to zero as $x$ approaches infinity. This behavior matches that of the inverse square function $f(x) = x^{-2}$.

• Graph B (Red): This graph shows a curve that begins at the origin and increases more steeply than linear but less steeply than cubic for $x > 0$. It represents the function $f(x) = x^{1/2}$, which is the square root function.

• Graph C (Blue): This graph looks similar to a linear function but is slightly curved, indicative of the cubic function's shape, which increases rapidly for large positive $x$ but less so than $x^5$. This corresponds to $f(x) = x^3$.

• Graph D (Green): This is a straight line passing through the origin with a slope of 1, which corresponds to the linear function $f(x) = x$.

• Graph E (Purple): This graph increases steeply for positive values of $x$ and shows a more pronounced steepness than the cubic graph. It represents the quintic function $f(x) = x^5$.

Matching Functions to Graphs:

• Graph A (Black): $f(x) = x^{-2}$
• Graph B (Red): $f(x) = x^{1/2}$
• Graph C (Blue): $f(x) = x^3$
• Graph D (Green): $f(x) = x$
• Graph E (Purple): $f(x) = x^5$

Final Matching:

• Graph A: $f(x) = x^{-2}$
• Graph B: $f(x) = x^{1/2}$
• Graph C: $f(x) = x^3$
• Graph D: $f(x) = x$
• Graph E: $f(x) = x^5$

Would you like further clarification or have any questions about these graphs and functions? Here are some additional areas we can discuss:

1. How do the degrees of the polynomial functions affect their shapes?
2. Why does the function $f(x) = x^{-2}$ behave differently from the others?
3. What are the general properties of odd and even power functions?
4. How can you distinguish between different power functions visually?
5. What are the key differences in behavior between $f(x) = x^3$ and $f(x) = x^5$?

Tip: When analyzing graphs of power functions, focus on the behavior as $x$ approaches 0 and as $x$ becomes very large (positive or negative). This helps in distinguishing between different types of power functions.