Question

PLEASE HELP! what can you say about the end behavior of the function f(x)=-4^6+6x^2-52? (picture of options attached-- please select all correct answers!)

Answer 1

The correct options are:

• f(x) is even,hence both the ends of f(x) goes in the same direction.

• The leading coefficient is negative, so the left end of the graph goes in downward direction.

Explanation:

We are given a polynomial function f(x) of degree 6 which is given by:

`f(x)=-4x^6+6x^2-52`

1)

Since the leading coefficient of the function f(x) is negative.

This means that the left end of the function will go in downward direction i.e. to negative infinity.

Since the function is an even polynomial function.

So, when

x → -∞ then -4 x^6 → -∞

Hence, option (1) is incorrect.

2)

Since f(x) is even hence both the ends of the graph in the same direction.

Hence, option (2) is incorrect.

3)

Option 3) is correct.

Both the ends of f(x) are in same direction since f(x) is even.

4)

Since the leading coefficient is negative and degree of the polynomial function is even.

Hence, the left end of the function goes downward direction.

Hence, option (4) is correct.

Answer 2

Answer: Option C and D are correct.

Explanation:

The end behavior of a function is the behavior of the graph of its graph as input value approaches positive infinity or negative infinity.

Also, The degree and the leading coefficient (The coefficient of the highest degree monomial) of a function shows the end behavior of its graph.

if Degree is even and leading coefficient is negative then the end behavior of the function f(x) is,

`f(x)\rightarrow -\infty` , as
`x\rightarrow -\infty`

`f(x)\rightarrow -\infty` , as
`x\rightarrow +\infty`

That is, Both ends of the graph goes down(In the same direction)

Here, the given function is,

`f(x) = -4x^6+6x^2-52`

Degree = 6(even), leading coefficient = -4 (negative)

Thus, the end behavior of the given function is,

Both ends of the graph goes down(In the same direction),

Hence, option C and D are correct.