Question

Consider j(x) = 12x^3/5 - 27x/5. Describe and provide an explanation for the end behavior of j(x).

Answer 1

To analyze the end behavior of the polynomial function \( j(x) = \frac{12}{5} x^3 - \frac{27}{5} x \), we need to focus on the leading term, which in this case is \( \frac{12}{5} x^3 \). The end behavior of a polynomial is largely determined by its highest-degree term, especially as \( x \) approaches positive or negative infinity.

### Step 1: Identify the leading term and its degree.
The leading term here is \( \frac{12}{5} x^3 \), and its degree is 3, which is odd.

### Step 2: Determine the coefficient of the leading term.
The coefficient of the leading term \( \frac{12}{5} \) is positive.

### Step 3: Analyze the end behavior based on degree and coefficient:

1. **As \( x \to +\infty \)** (positive infinity):
- Since the leading term is \( \frac{12}{5} x^3 \) and its degree is odd with a positive coefficient, \( j(x) \) will approach \( +\infty \).
- Therefore, \( j(x) \to +\infty \) as \( x \to +\infty \).

2. **As \( x \to -\infty \)** (negative infinity):
- The same leading term will dominate, but since the degree is odd, \( j(x) \) will approach \( -\infty \).
- Therefore, \( j(x) \to -\infty \) as \( x \to -\infty \).

### Conclusion:
In summary, the end behavior of the polynomial function \( j(x) \) is:
- As \( x \to +\infty, j(x) \to +\infty \)
- As \( x \to -\infty, j(x) \to -\infty \)

This indicates that the graph of \( j(x) \) will rise to infinity on the right side (as \( x \) becomes very large) and fall to negative infinity on the left side (as \( x \) becomes very negative).