Problem 22 - College Algebra

Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$ f(x)=11 x^{4}-6 x^{2}+x+3 $$

Step-by-Step Solution

Answer

The end behavior of the graph of the given polynomial function is: as \(x \to \infty, f(x) \to \infty\) and as \(x \to -\infty, f(x) \to \infty\)

Step 1: Identifying degree and leading coefficient

For the function \(f(x)= 11 x^{4} - 6 x^{2} + x + 3\), identify the highest power of x and its corresponding coefficient. Here, 4 is the highest power hence 4 is the degree and 11 is the leading coefficient.

Step 2: Applying the Leading Coefficient Test

Since the degree is even and leading coefficient is positive, the end behavior will be as follows: as \(x \to \infty, f(x) \to \infty\) and as \(x \to -\infty, f(x) \to \infty\)