2.2 sections of work

Question

Describe the end behavior of the polynomial functions f

1.) f(x) = 4x⁴ + 2x³ =1

2.) f(x) = (x+2)² (2x-1)

Answer 1

In end behavior, the only term that you need to consider is the one with the highest exponent. The other terms influence the graph near the origin, but not when x is extremely positive or extremely negative.

So for part a of your problem you need only look at the x⁴ term. You can ignore the coefficient 4.

When x is a large positive number, x⁴ is very positive. so the graph is rising to the right.

When x is a large negative number, x⁴ is still positive, so the graph is also rising to the left.

So for part b of your problem you need only consider the largest exponent when it is expanded.

The squared binomial will multiply out to x². Then it is multiplied by another binomial with an x.

So the power is x³.

When x is a large positive number, x³ is positive, so the graph is rising to the right.

When x is a large negative number, x³ is negative, so the graph is falling to the left.

Answer 2

Jame,

Probably, the 1st question must be f(x) = 4x⁴ + 2x³ -1 (or f(x) = 4x⁴ + 2x³ +1) - but that would not change the answer. I'd use f(x) = 4x⁴ + 2x³ -1

1) f(x) = 4x⁴ + 2x³ -1.

when x---> + infinity or - infinity, the term 4x⁴ would be dominant I4x⁴I/I2x³I=2IxI>>1

so f(x) --->I4x⁴I

2.) f(x) = (x+2)² (2x-1)

when x---> + infinity or - infinity (x+2)² would behave like x² so f(x) ---> x² _* 2x=2x³

Hope it'd help. Good luck

Dr.Ariel B.