3.11.3 Quiz: Comparing and Analyzing Function Types
Question 3 of 10
What can you say about the end behavior of the function
?
A. The leading coefficient is positive so the left end of the grapt
goes down.
B. The leading coefficient is positive so the left end of the grapt
goes up. is an even function so both ends of the graph go in the s
direction.
D. is an even function so both ends of the graph go in oppe
directions.

Ask by Macdonald Rowe. in the United States

Nov 07,2024

Question

3.11.3 Quiz: Comparing and Analyzing Function Types
Question 3 of 10
What can you say about the end behavior of the function
?
A. The leading coefficient is positive so the left end of the grapt
goes down.
B. The leading coefficient is positive so the left end of the grapt
goes up. is an even function so both ends of the graph go in the s
direction.
D. is an even function so both ends of the graph go in oppe
directions.

Ask by Macdonald Rowe. in the United States

Nov 07,2024

UpStudy AI · Accepted Answer

To determine the end behavior of the function $$ f(x) = -4x^6 + 6x^2 - 52 $$, we need to consider the leading term of the function and the degree of the polynomial.

The leading term of the function is the term with the highest degree, which in this case is $$ -4x^6 $$. The leading coefficient is the coefficient of the leading term, which is -4.

The degree of the polynomial is the highest power of the variable, which in this case is 6.

Now let's analyze the end behavior based on the leading coefficient and the degree:

1. If the leading coefficient is positive, the graph will go up as $$ x $$ approaches positive infinity and down as $$ x $$ approaches negative infinity.
2. If the leading coefficient is negative, the graph will go down as $$ x $$ approaches positive infinity and up as $$ x $$ approaches negative infinity.
3. If the degree of the polynomial is even, the graph will approach the same direction (either up or down) as $$ x $$ approaches positive or negative infinity.
4. If the degree of the polynomial is odd, the graph will approach opposite directions as $$ x $$ approaches positive or negative infinity.

Given that the leading coefficient is -4 (negative) and the degree is 6 (even), the end behavior of the function $$ f(x) $$ is as follows:

- As $$ x $$ approaches positive infinity, $$ f(x) $$ will go down because the leading coefficient is negative.
- As $$ x $$ approaches negative infinity, $$ f(x) $$ will go up because the leading coefficient is negative.

Therefore, the correct answer is:

A. The leading coefficient is positive so the left end of the graph goes down.

However, there is a mistake in the provided options. The correct statement should be:

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

None of the provided options correctly describe the end behavior of the function.
The function $$ f(x) = -4x^6 + 6x^2 - 52 $$ has a negative leading coefficient and an even degree. Thus, as $$ x $$ approaches positive infinity, $$ f(x) $$ goes down, and as $$ x $$ approaches negative infinity, $$ f(x) $$ goes up.

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