Problem 21 - College Algebra

Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$ f(x)=5 x^{4}+7 x^{2}-x+9 $$

Step-by-Step Solution

Answer

The end behavior of the given polynomial function is 'Up to the right and up to the left'.

Step 1: Identify the Degree and Leading Coefficient

The given polynomial function is \(f(x) = 5x^{4} + 7x^{2} - x + 9\). Here, the highest power of variable \(x\) is 4, hence the degree is 4. The coefficient of the term with highest degree is 5, hence the leading coefficient is 5.

Step 2: Apply the Leading Coefficient Test

Since the degree is even and the leading coefficient is positive, according to the Leading Coefficient Test, the function rises to the right and rises to the left (meaning As \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity and as \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity as well.

Step 3: State the end behavior

From Step 2, we can observe that as \(x\) approaches positive or negative infinity, \(f(x)\) approaches positive infinity. Therefore, the end behavior of the polynomial can be modeled as: 'Up to the right and up to the left'