Answer :
To determine the end behavior of the polynomial function [tex]\( f(x) = -5x^4 + 3x^2 - 7x + 8 \)[/tex] using the Leading Coefficient Test, follow these steps:
1. Identify the leading term: The leading term is the term with the highest power of [tex]\( x \)[/tex]. In this polynomial, the leading term is [tex]\( -5x^4 \)[/tex].
2. Determine the degree of the polynomial: The degree of the polynomial is the highest power of [tex]\( x \)[/tex] present in the polynomial, which is 4.
3. Determine the leading coefficient: The leading coefficient is the coefficient of the leading term. In this case, the leading coefficient is -5.
4. Evaluate the end behavior based on the degree and the leading coefficient:
- If the degree is even and the leading coefficient is positive, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex].
- If the degree is even and the leading coefficient is negative, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- If the degree is odd and the leading coefficient is positive, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- If the degree is odd and the leading coefficient is negative, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex].
Since the degree of the polynomial [tex]\( f(x) = -5x^4 + 3x^2 - 7x + 8 \)[/tex] is 4 (even) and the leading coefficient is -5 (negative), the end behavior of the graph is:
- As [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] (the graph falls to the right).
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] (the graph falls to the left).
Therefore, the correct answer is:
B. The graph of [tex]\( f(x) \)[/tex] falls to the left and falls to the right.
1. Identify the leading term: The leading term is the term with the highest power of [tex]\( x \)[/tex]. In this polynomial, the leading term is [tex]\( -5x^4 \)[/tex].
2. Determine the degree of the polynomial: The degree of the polynomial is the highest power of [tex]\( x \)[/tex] present in the polynomial, which is 4.
3. Determine the leading coefficient: The leading coefficient is the coefficient of the leading term. In this case, the leading coefficient is -5.
4. Evaluate the end behavior based on the degree and the leading coefficient:
- If the degree is even and the leading coefficient is positive, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex].
- If the degree is even and the leading coefficient is negative, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- If the degree is odd and the leading coefficient is positive, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- If the degree is odd and the leading coefficient is negative, then as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex], and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex].
Since the degree of the polynomial [tex]\( f(x) = -5x^4 + 3x^2 - 7x + 8 \)[/tex] is 4 (even) and the leading coefficient is -5 (negative), the end behavior of the graph is:
- As [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] (the graph falls to the right).
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] (the graph falls to the left).
Therefore, the correct answer is:
B. The graph of [tex]\( f(x) \)[/tex] falls to the left and falls to the right.
