To determine the end behavior of the polynomial function [tex]\( f(x) = 8x^4 + 7x^2 - x + 5 \)[/tex], we will use the Leading Coefficient Test, which involves looking at the degree of the polynomial and the sign of the leading coefficient.
Here is a detailed, step-by-step solution:
1. Identify the degree of the polynomial:
- The degree of a polynomial is the highest power of [tex]\(x\)[/tex] in the polynomial. In this case, the highest power of [tex]\(x\)[/tex] is [tex]\(x^4\)[/tex].
- Therefore, the degree of the polynomial [tex]\( f(x) = 8x^4 + 7x^2 - x + 5 \)[/tex] is 4.
2. Identify the leading coefficient:
- The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex]. In the given polynomial, the term with the highest power is [tex]\(8x^4\)[/tex], and the leading coefficient is 8.
3. Determine the end behavior based on the leading coefficient and the degree:
- When the degree of the polynomial is even and the leading coefficient is positive, the end behavior of the polynomial is such that the graph rises to the left and rises to the right.
Given the above information:
- The degree of 4 is even.
- The leading coefficient 8 is positive.
Thus, based on the Leading Coefficient Test, the correct determination of the end behavior is:
- The graph of [tex]\( f(x) = 8x^4 + 7x^2 - x + 5 \)[/tex] rises to the left and rises to the right.
Therefore, the correct answer is:
D. The graph of [tex]\(f(x)\)[/tex] rises to the left and rises to the right.
