To analyze the end behavior of the polynomial function
\[ f(x) = 3.18x^4 + 6x^2 + x - 6, \]
we need to look at the leading term of the polynomial, as this term will dominate the function's behavior for very large and very small values of \(x\).
-
Identify the leading term: The leading term in this function is \(3.18x^4\), which is the term with the highest exponent (degree 4).
-
Analyze the leading coefficient and degree:
- The degree of the polynomial is 4 (an even degree).
- The leading coefficient (3.18) is positive.
-
Determine end behavior:
- For an even degree polynomial with a positive leading coefficient, the end behavior is as follows:
- As \(x \to +\infty\) (as \(x\) approaches positive infinity), \(f(x) \to +\infty\) (the graph will rise).
- As \(x \to -\infty\) (as \(x\) approaches negative infinity), \(f(x) \to +\infty\) (the graph will also rise).
- For an even degree polynomial with a positive leading coefficient, the end behavior is as follows:
In conclusion, the end behavior of the function \(f(x)\) is:
- \(f(x) \to +\infty\) as \(x \to +\infty\)
- \(f(x) \to +\infty\) as \(x \to -\infty\)
The graph will rise to the right and rise to the left.
1 answer