Answer :
To determine the end behavior of the function \( f(x) = -4x^6 + 6x^2 - 52 \), follow these detailed steps:
1. Identify the leading term:
- The leading term in a polynomial is the term with the highest power of \( x \). In this case, the leading term is \( -4x^6 \).
2. Analyze the degree of the polynomial:
- The degree of a polynomial is the highest power of \( x \) present in the polynomial. Here, the degree is 6, which is an even number.
3. Determine the sign of the leading coefficient:
- The leading coefficient is the coefficient of the leading term. For \( -4x^6 \), the leading coefficient is \(-4\), which is negative.
4. Determine the end behavior based on the degree and leading coefficient:
- Since the degree of the polynomial is even, both ends of the graph of the polynomial will go in the same direction.
- Since the leading coefficient is negative, both ends of the graph will go down.
5. Conclusion on the end behavior:
- With an even degree and a negative leading coefficient, the graph of the polynomial \( f(x) = -4x^6 + 6x^2 - 52 \) will have both ends going down as \( x \to \infty \) and \( x \to -\infty \).
Ensuring all details align with our understanding of polynomial graph behaviors, we find the correct description of the end behavior. Therefore, the correct statement about \( f(x) \) is:
[tex]\[ \boxed{B. \, f(x) \, \text{is an even function so both ends of the graph go in the same direction.}} \][/tex]
1. Identify the leading term:
- The leading term in a polynomial is the term with the highest power of \( x \). In this case, the leading term is \( -4x^6 \).
2. Analyze the degree of the polynomial:
- The degree of a polynomial is the highest power of \( x \) present in the polynomial. Here, the degree is 6, which is an even number.
3. Determine the sign of the leading coefficient:
- The leading coefficient is the coefficient of the leading term. For \( -4x^6 \), the leading coefficient is \(-4\), which is negative.
4. Determine the end behavior based on the degree and leading coefficient:
- Since the degree of the polynomial is even, both ends of the graph of the polynomial will go in the same direction.
- Since the leading coefficient is negative, both ends of the graph will go down.
5. Conclusion on the end behavior:
- With an even degree and a negative leading coefficient, the graph of the polynomial \( f(x) = -4x^6 + 6x^2 - 52 \) will have both ends going down as \( x \to \infty \) and \( x \to -\infty \).
Ensuring all details align with our understanding of polynomial graph behaviors, we find the correct description of the end behavior. Therefore, the correct statement about \( f(x) \) is:
[tex]\[ \boxed{B. \, f(x) \, \text{is an even function so both ends of the graph go in the same direction.}} \][/tex]
