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
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.
Yes, I can help you understand this problem. To determine which polynomials could represent the given end behavior, we need to consider the leading term of each polynomial.
The leading term of a polynomial is the term with the highest power of x. For example, in the polynomial -5x^6 + 10x^5 - 3x^2 + 9, the leading term is -5x^6.
If the leading term has an even degree (e.g., x^2, x^4, etc.), then as x approaches infinity or negative infinity, the function will always approach positive infinity because any positive number squared or raised to an even power is positive. Therefore, polynomials with even degree leading terms cannot represent the given end behavior.
On the other hand, if the leading term has an odd degree (e.g., x^3, x^5, etc.), then as x approaches infinity, the function will approach positive infinity, and as x approaches negative infinity, the function will approach negative infinity. This is because any positive number cubed or raised to an odd power remains positive, while any negative number cubed or raised to an odd power remains negative. Therefore, polynomials with odd degree leading terms could represent the given end behavior.
Let's analyze each polynomial:
a. -x^2: This polynomial has an even degree leading term, so it cannot represent the given end behavior.
b. 2x^4 - x^3 - x^2 - x - 1: This polynomial has an even degree leading term, so it cannot represent the given end behavior.
c. -5x^6 + 10x^5 - 3x^2 + 9: This polynomial has an even degree leading term, so it cannot represent the given end behavior.
d. x^2: This polynomial has an even degree leading term, so it cannot represent the given end behavior.
e. -2x^3 + 16x: This polynomial has an odd degree leading term, so it can represent the given end behavior.
Therefore, the correct answer is e. -2x^3 + 16x.