Respuesta :
Answer: In summary, the function \(f(x) = (-2(x-1)^3)(x+2)^2\) exhibits this end behavior.
Step-by-step explanation:
To determine the **end behavior** of the function \(f(x) = (-2(x-1)^3)(x+2)^2\), let's analyze its leading term. The leading term is the one with the highest exponent.
1. **Leading Term**:
- The expression \((-2(x-1)^3)(x+2)^2\) has two factors: \(-2(x-1)^3\) and \((x+2)^2\).
- The leading term will be the product of the highest power terms from each factor.
- The highest power term in \(-2(x-1)^3\) is \(-2(x-1)^3\), which has a degree of 3.
- The highest power term in \((x+2)^2\) is \((x+2)^2\), which has a degree of 2.
- Therefore, the leading term of the entire function is \(-2(x-1)^3(x+2)^2\).
2. **End Behavior**:
- Since the leading term has a negative coefficient (\(-2\)), the function will behave differently depending on whether the degree of the polynomial is even or odd.
- If the degree is **even**, the end behavior will be similar to that of a quadratic function. As \(x\) approaches positive or negative infinity, the function will approach **negative infinity**.
- If the degree is **odd**, the end behavior will be similar to that of a cubic function. As \(x\) approaches positive infinity, the function will approach **negative infinity**, and as \(x\) approaches negative infinity, the function will approach **positive infinity**.
3. **Conclusion**:
- Since the degree of the leading term is **odd** (3), the end behavior of the function \(f(x)\) is as follows:
- As \(x\) approaches positive infinity, \(f(x)\) approaches **negative infinity**.
- As \(x\) approaches negative infinity, \(f(x)\) approaches **positive infinity**.
In summary, the function \(f(x) = (-2(x-1)^3)(x+2^2\) exhibits this end behavior.
