To solve this problem, we need to categorize each function and graph based on their end behavior.

End Behavior Categories:

1. Category 1:
   $\lim_{{x \to -\infty}} f(x) = +\infty$ and $\lim_{{x \to \infty}} f(x) = +\infty$
   Description: As $x$ goes to $-\infty$ or $+\infty$, the function goes to $+\infty$.

2. Category 2:
   $\lim_{{x \to -\infty}} f(x) = -\infty$ and $\lim_{{x \to \infty}} f(x) = -\infty$
   Description: As $x$ goes to $-\infty$ or $+\infty$, the function goes to $-\infty$.

3. Category 3:
   $\lim_{{x \to -\infty}} f(x) = -\infty$ and $\lim_{{x \to \infty}} f(x) = +\infty$
   Description: As $x$ goes to $-\infty$, the function goes to $-\infty$ and as $x$ goes to $+\infty$, it goes to $+\infty$.

4. Category 4:
   $\lim_{{x \to -\infty}} f(x) = +\infty$ and $\lim_{{x \to \infty}} f(x) = -\infty$
   Description: As $x$ goes to $-\infty$, the function goes to $+\infty$ and as $x$ goes to $+\infty$, it goes to $-\infty$.

Analyzing the Functions:

To determine the end behavior of each function, we analyze the degree and leading coefficient of each polynomial:

1. $f(x) = x^4 + \cdots$:

   • Degree: 4 (even)
   • Leading coefficient: Positive
   • End Behavior: As $x \to \pm\infty$, $f(x) \to +\infty$.
   • Category 1: Both ends go to $+\infty$.

2. $f(x) = 4x - 3x^6 + 5$:

   • Degree: 6 (even)
   • Leading coefficient: Negative
   • End Behavior: As $x \to \pm\infty$, $f(x) \to -\infty$.
   • Category 2: Both ends go to $-\infty$.

3. $f(x) = -2x^3 + 4x - 7$:

   • Degree: 3 (odd)
   • Leading coefficient: Negative
   • End Behavior: As $x \to -\infty$, $f(x) \to +\infty$; as $x \to \infty$, $f(x) \to -\infty$.
   • Category 4: One end goes to $+\infty$, the other to $-\infty$.

4. $f(x) = 3(x - 2)^2(x + 1)^3$:

   • Degree: 5 (odd)
   • Leading coefficient: Positive
   • End Behavior: As $x \to -\infty$, $f(x) \to -\infty$; as $x \to \infty$, $f(x) \to +\infty$.
   • Category 3: One end goes to $-\infty$, the other to $+\infty$.

5. $f(x) = 9 - x^2(x^2 + 4x + 4)$:

   • Degree: 4 (even)
   • Leading coefficient: Negative (after expansion)
   • End Behavior: As $x \to \pm\infty$, $f(x) \to -\infty$.
   • Category 2: Both ends go to $-\infty$.

6. $f(x) = -x^2(x - 1)(x + 4)$:

   • Degree: 4 (even)
   • Leading coefficient: Negative
   • End Behavior: As $x \to \pm\infty$, $f(x) \to -\infty$.
   • Category 2: Both ends go to $-\infty$.

Matching Graphs:

• Analyze the graphs to match them with the correct categories based on how the ends of each graph behave (going up or down).

Would you like me to continue matching the graphs and providing more detailed explanations?