Consider the polynomial function $$ f(t)=t^{9}+2 t^{6}+3 t^{2}+4 $$ As $$ t \rightarrow-\infty, f(t) \rightarrow $$ ? As $$ t \rightarrow \infty, f(t) \rightarrow $$ ?

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UpStudy AI · Accepted Answer

The polynomial function is 
$$
f(t)=t^9+2t^6+3t^2+4.
$$

**Step 1: Identify the leading term.**

The highest degree term is $$ t^9 $$ which determines the end behavior of the function since its power $$ 9 $$ is odd.

**Step 2: Analyze the behavior as $$ t \rightarrow \infty $$.**

When $$ t $$ is very large and positive:
- $$ t^9 $$ becomes very large and positive.
- The lower degree terms $$2t^6$$, $$3t^2$$, and $$4$$ are much smaller in magnitude compared to $$ t^9 $$.

Therefore, 
$$
f(t) \approx t^9 \quad \text{as} \quad t \rightarrow \infty,
$$
which means 
$$
f(t) \rightarrow \infty.
$$

**Step 3: Analyze the behavior as $$ t \rightarrow -\infty $$.**

When $$ t $$ is very large in magnitude but negative:
- $$ t^9 $$ will be:

$$
t^9 = (-|t|)^9 = -|t|^9,
$$
which is very large and negative.

- The other terms $$2t^6$$, $$3t^2$$, and $$4$$ still remain positive or small relative to the magnitude of $$ t^9 $$ (with even exponents producing positive values) but they do not counteract the negative large magnitude of $$ t^9 $$.

Thus,
$$
f(t) \approx t^9 \quad \text{as} \quad t \rightarrow -\infty,
$$
which means 
$$
f(t) \rightarrow -\infty.
$$

**Final Answers:**

$$
\text{As } t \rightarrow -\infty, f(t) \rightarrow -\infty.
$$
$$
\text{As } t \rightarrow \infty, f(t) \rightarrow \infty.
$$
As $$ t $$ approaches negative infinity, $$ f(t) $$ approaches negative infinity. As $$ t $$ approaches positive infinity, $$ f(t) $$ approaches positive infinity.

Consider the polynomial function \( f(t)=t^{9}+2 t^{6}+3 t^{2}+4 \) As \( t \rightarrow-\infty, f(t) \rightarrow \) ? As \( t \rightarrow \infty, f(t) \rightarrow \) ?

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