What can you say about the end behavior of the function [tex]f(x)=-4x^6+6x^2-52?[/tex]

A. [tex]f(x)[/tex] is an even function, so both ends of the graph go in opposite directions.

B. The leading coefficient is negative, so the left end of the graph goes down.

C. The leading coefficient is negative, so the left end of the graph goes up.

D. [tex]f(x)[/tex] is an even function, so both ends of the graph go in the same direction.



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.