Question

Suppose the polynomial f(x) has the following end behavior: as x→∞, f(x)→−∞, and as x→−∞, f(x)→−∞.

Which of the following polynomials could represent f(x)?

There may be more than one correct answer. Select all correct answers.

a. −x2


b. 2x^4−x^3−x^2−x−1

c. −5x^6+10x^5−3x^2+9

d. x^2

e. −2x^3+16x


can someone please help me I dont understand this

Answer 1

Is it the notation x→∞, f(x)→−∞ that you don't understand
In words in means: as x gets bigger and bigger, the y drops further and further down below the x-axis
that is, into quadrant IV

for each equation, you only have to consider the term with the highest exponent.
as x becomes larger, the higher exponent term will dominate and the other terms become less and less significant.
e.g. y = 2x^4−x^3−x^2−x−1
suppose x = 1000, really not that big yet.
y = 2(1,000,000,000,000)- 1,000,000,000 - 1000000 - 1000 - 1
notice the real value of the number comes from the 1st term

clearly that can't be the one we are looking for, since the results will always be positive for large values of x
Also, it has to be an even exponent and a negative result, so
it has to be a) or c)

It can't be the leading odd ones, since a large value of x would yield a negative
and a large negative x would yield a positive y.