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The function given is:

\[
g(t) = -\frac{2}{9}t^4 + 3t
\]

### Left-hand and right-hand behavior:

1. **As \( t \to -\infty \) (left-hand behavior):**
   - The highest degree term dominates the behavior of the function as \( t \to -\infty \). The term \( -\frac{2}{9}t^4 \) grows very large negatively, since \( t^4 \) will grow large and positive, but the negative sign in front makes the whole term negative.
   - The linear term \( 3t \) contributes a smaller effect because \( t^4 \) dominates. Therefore, as \( t \to -\infty \), \( g(t) \to -\infty \).

2. **As \( t \to \infty \) (right-hand behavior):**
   - Similarly, for \( t \to \infty \), the term \( -\frac{2}{9}t^4 \) dominates. Since \( t^4 \) becomes very large and the negative coefficient makes it negative, the function will tend to \( -\infty \) as well.
   - The linear term \( 3t \) again has a smaller influence in comparison to the dominant \( t^4 \)-term. Thus, as \( t \to \infty \), \( g(t) \to -\infty \).

### Critical points:
To better understand the behavior and find any critical points, we differentiate \( g(t) \) with respect to \( t \):

\[
g'(t) = -\frac{8}{9}t^3 + 3
\]

Setting \( g'(t) = 0 \) to find critical points:

\[
-\frac{8}{9}t^3 + 3 = 0
\]
\[
t^3 = \frac{27}{8}
\]
\[
t = \sqrt[3]{\frac{27}{8}} = \frac{3}{2}
\]

So, \( t = \frac{3}{2} \) is a critical point. We can analyze the second derivative to classify the critical point as a local maximum or minimum.

\[
g''(t) = -\frac{24}{9}t^2 = -\frac{8}{3}t^2
\]

At \( t = \frac{3}{2} \):

\[
g''\left( \frac{3}{2} \right) = -\frac{8}{3} \times \left( \frac{3}{2} \right)^2 = -6
\]

Since \( g''(t) < 0 \), the function has a local maximum at \( t = \frac{3}{2} \).

### Sketch of the graph:
- The function has a local maximum at \( t = \frac{3}{2} \), and the value of \( g(t) \) at this point can be found by substituting \( t = \frac{3}{2} \) into the original function.
- The graph falls towards \( -\infty \) on both sides (left and right behavior), with the local maximum in between.

Let me know if you'd like me to generate the graph.

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### Relative questions:
1. What is the significance of critical points in understanding a function's graph?
2. How does the degree of a polynomial affect its end behavior?
3. What is the importance of the second derivative in determining concavity?
4. Can a function have more than one local maximum or minimum?
5. How do inflection points relate to the curvature of a graph?

**Tip:** When analyzing polynomials, the highest degree term typically dictates the end behavior of the graph.

Describe the left-hand and right-hand behavior of the graph of the function, then sketch its graph. g(t) = -(2/9)t^4 + 3t

Learn how to describe the left-hand and right-hand behavior of the polynomial function g(t) = -(2/9)t^4 + 3t. This problem explores key mathematical concepts, such as the derivative, second derivative, end behavior, and critical points. Suitable for students in Grades 10-12.

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