To determine the end behavior of the polynomial function [tex]\( f(x) = 4x^3 + 2x^2 - 2x - 2 \)[/tex] using the Leading Coefficient Test, we need to follow these steps:
1. Identify the Leading Coefficient and the Degree of the Polynomial:
The leading term of the polynomial is the term with the highest power of [tex]\( x \)[/tex]. In this function, the leading term is [tex]\( 4x^3 \)[/tex].
- The leading coefficient is the coefficient of the leading term, which is [tex]\( 4 \)[/tex].
- The degree of the polynomial is the highest power of [tex]\( x \)[/tex], which is [tex]\( 3 \)[/tex].
2. Determine the End Behavior Based on the Leading Coefficient and Degree:
The Leading Coefficient Test provides us with rules based on the leading coefficient and the degree of the polynomial to determine the end behavior:
- If the degree is odd:
- And the leading coefficient is positive: The polynomial will rise to the right and fall to the left.
- And the leading coefficient is negative: The polynomial will fall to the right and rise to the left.
- If the degree is even:
- And the leading coefficient is positive: The polynomial will rise to the left and to the right.
- And the leading coefficient is negative: The polynomial will fall to the left and to the right.
3. Apply the Leading Coefficient Test to Our Polynomial:
- The degree of our polynomial [tex]\( 4x^3 + 2x^2 - 2x - 2 \)[/tex] is [tex]\( 3 \)[/tex], which is odd.
- The leading coefficient is [tex]\( 4 \)[/tex], which is positive.
According to the rules of the Leading Coefficient Test:
- Since the degree is odd and the leading coefficient is positive, the polynomial will rise to the right and fall to the left.
Hence, the end behavior of the polynomial function [tex]\( f(x) = 4x^3 + 2x^2 - 2x - 2 \)[/tex] is as follows:
- As [tex]\( x \to \infty \)[/tex] (as [tex]\( x \)[/tex] approaches positive infinity), [tex]\( f(x) \)[/tex] [tex]\(\to \infty\)[/tex].
- As [tex]\( x \to -\infty \)[/tex] (as [tex]\( x \)[/tex] approaches negative infinity), [tex]\( f(x) \)[/tex] [tex]\(\to -\infty\)[/tex].
In conclusion, the polynomial function [tex]\( f(x) = 4x^3 + 2x^2 - 2x - 2 \)[/tex] rises to the right and falls to the left. This end behavior helps us match the function with its graph.
