Let  
f
(
x
)
=
2
x
+
4
x
2
−
2
x
 . Use the leading terms to explain what will happen to the function values as  
x
  gets larger and larger. Check your answer using a table or a graph.

Answer: As  
x
  gets larger, the function values get closer to

To analyze the behavior of the function \( f(x) = 2x + 4x^2 - 2x \) as \( x \) gets larger, we can examine the leading term, which is the term with the highest degree in \( x \). This approach helps us understand how \( f(x) \) behaves as \( x \to \infty \).

### Step 1: Identify the Leading Term
The function \( f(x) \) can be simplified to:
\[
f(x) = 4x^2
\]
since the linear terms \( 2x \) and \(-2x\) cancel each other out. 

Thus, the leading term is \( 4x^2 \), which dominates the function’s behavior for large values of \( x \).

### Step 2: Analyze the Leading Term as \( x \to \infty \)
As \( x \) grows larger, \( 4x^2 \) will also grow larger in a positive direction, since \( 4x^2 \) is positive for all values of \( x \). Therefore, as \( x \to \infty \), the values of \( f(x) \) will also tend toward infinity.

### Conclusion
As \( x \) gets larger, the function values \( f(x) \) will increase without bound, getting closer to \( +\infty \).

Would you like me to verify this result using a table of values or a graph for further confirmation?

---

Here are 5 related questions to expand your understanding:

1. How would the function behave as \( x \to -\infty \)?
2. What is the effect of adding a constant to \( f(x) \) on the end behavior of the function?
3. How do you determine which terms in a polynomial dominate as \( x \to \infty \)?
4. If the coefficient of \( x^2 \) was negative, how would the function behave as \( x \to \infty \)?
5. What role do non-leading terms play in determining the function’s behavior for small values of \( x \)?

**Tip:** For large values of \( x \), focus on the highest degree term in a polynomial, as it usually dictates the function's end behavior.

Let f(x) = 2x + 4x^2 - 2x. Use the leading terms to explain what will happen to the function values as x gets larger and larger. Check your answer using a table or a graph.

Understand how to analyze the end behavior of the polynomial function f(x) = 2x + 4x^2 - 2x by examining the leading term. Discover how the function behaves as x approaches infinity, with a detailed explanation suitable for high school students in Grades 10-12.

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f ( x )

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