Answer :
To determine the end behavior of the function \( f(x) = -4x^6 + 6x^2 - 52 \), let’s follow these steps:
### Step 1: Identify the Degree and Leading Coefficient
1. Degree of the function: The highest power of \( x \) in the polynomial \( f(x) \) is \( x^6 \). Hence, the degree of the function is 6.
2. Leading coefficient: The coefficient of the term with the highest degree (which is \( x^6 \) in this case) is \(-4\).
### Step 2: Analyze the Degree
Since the degree of the polynomial is 6 (which is an even number), we know that the end behavior will be such that both ends of the graph go in the same direction.
### Step 3: Analyze the Leading Coefficient
The leading coefficient is \(-4\), which is negative.
### Step 4: Combining the Information
With an even degree and a negative leading coefficient:
- Both ends of the graph will go in the same direction.
- Since the leading coefficient is negative, as \( x \) approaches \( \infty \) or \( -\infty \), the value of \( f(x) \) will go to \(-\infty \). Therefore, both ends of the graph will go downward.
### Conclusion
Based on the analysis:
- The polynomial is of even degree, so both ends go in the same direction.
- The leading coefficient is negative, so both ends go downwards.
Thus, the correct answer is:
D. [tex]\( f(x) \)[/tex] is an even function so both ends of the graph go in the same direction.
### Step 1: Identify the Degree and Leading Coefficient
1. Degree of the function: The highest power of \( x \) in the polynomial \( f(x) \) is \( x^6 \). Hence, the degree of the function is 6.
2. Leading coefficient: The coefficient of the term with the highest degree (which is \( x^6 \) in this case) is \(-4\).
### Step 2: Analyze the Degree
Since the degree of the polynomial is 6 (which is an even number), we know that the end behavior will be such that both ends of the graph go in the same direction.
### Step 3: Analyze the Leading Coefficient
The leading coefficient is \(-4\), which is negative.
### Step 4: Combining the Information
With an even degree and a negative leading coefficient:
- Both ends of the graph will go in the same direction.
- Since the leading coefficient is negative, as \( x \) approaches \( \infty \) or \( -\infty \), the value of \( f(x) \) will go to \(-\infty \). Therefore, both ends of the graph will go downward.
### Conclusion
Based on the analysis:
- The polynomial is of even degree, so both ends go in the same direction.
- The leading coefficient is negative, so both ends go downwards.
Thus, the correct answer is:
D. [tex]\( f(x) \)[/tex] is an even function so both ends of the graph go in the same direction.
