How do you find the end behavior of #9x^4 - 8x^3 + 4x#?

Question

Liam Brennan · Accepted Answer

As #x##rarr##oo# #y##rarr##oo# . As #x# #rarr#-#oo#, #y# #rarr# #oo#.

graph{9x^4-8x^3+4x [-9.625, 10.375, -2.4, 7.6]} Finding the end behavior is finding what happens as x goes to positive and negative infinity. On the graph, look at the two directions the graph is going: left and right (x) and up and down (y). First lets look at what happens when x goes to positive infinity, the graph is going right (towards positive infinity) and most importantly the graph is also going up (the y values are increasing). This gives the first part of the answer. Now lets look at the negative side the x values are decreasing but the y values are increasing. This tells us that if you keep putting in negative values the y values will keep going up, hence the second part of the answer.

Sadie Ackerman · Answer

undefined To find the end behavior of the polynomial $9x^4 - 8x^3 + 4x$, you need to look at the leading term, which is $9x^4$. The end behavior of a polynomial is determined by the degree and the leading coefficient of the polynomial.

Since the leading term $9x^4$ is a fourth-degree term with a positive leading coefficient $9$, the end behavior of the polynomial is as follows:

- As $x$ approaches positive infinity ($+\infty$), the polynomial increases without bound.
- As $x$ approaches negative infinity ($-\infty$), the polynomial decreases without bound.

Therefore, the end behavior of $9x^4 - 8x^3 + 4x$ is that it rises on the right and falls on the left.