Question

. Consider the leading term of the polynomial function. What is the end behavior of the graph? Describe the end behavior and provide the leading term.

-3x5 + 9x4 + 5x3 + 3

Answer 1

-15+36+15+3=

-15+15=0=

36+ 3=

39

Answer 2

The end behaviour of this function is -ꝏ when x approaches +ꝏ and +ꝏ when x approaches -ꝏ. The leading term is -3x^5.

Explanation:

The end behaviour can be found by taking the limits of function for x approaching +ꝏ and -ꝏ:

lim x→+/-ꝏ (-3x^5+9x^4+5x^3+3)

To solve the limits it must be factorized the highest term in the polynomial:

lim x→+/-ꝏ {-3x^5*[-3x^5/(-3x^5)+9x^4/(-3x^5)+5x^3/(-3x^5)+3/(-3x^5)]}

By simplifying:

lim x→+/-ꝏ {-3x^5*[1-3/x-5/3x^2-1/x^5)]}

If we first take the limits in the within the square brackets, we will notice than whether the limits approaches +ꝏ or -ꝏ the limits of each of these terms tend to zero due to the denominator is greater than the numerator:

lim x→+/-ꝏ {-3x^5*[1-0-0-0)]}

lim x→+/-ꝏ {-3x^5*[1]}

lim x→+/-ꝏ {-3x^5}

This way we find the leading term. Now taking the positive and negative limits:

lim x→+ꝏ {-3x^5} = -ꝏ

and:

lim x→-ꝏ {-3x^5} = +ꝏ