What is the end behavior of the graph of the polynomial function f(x) = -x⁵ + 9x⁴ - 18x³?

Solution:

The end behavior of a polynomial function implies that how f(x) behaves when x approaches infinity on both sides of the number line i.e. -∞ and ∞. This could be ascertained easily by visualizing the given polynomial equation on a graph. The graph of the equation f(x) = -x⁵ + 9x⁴ - 18x³ is given below:

If we look at the upper end of the graph i.e. (y = f(x), - 6, 23328), it can be seen that it is moving upwards(north direction). As the value of ‘x’ increases towards the left-hand side of the graph moves further upwards. So we conclude that as,

x → -∞, f(x) → ∞ --- (1)

Now let us look at the lower end of the graph i.e. (y = f(x), 10, - 28000), it can be seen that it is moving downwards (southwest direction). As the value of x decreases (i.e. x → -∞), the graph of the equation moves further downwards. Hence we can infer that:

x → ∞, f(x) → -∞ --- (2)

Thus equation (1) and (2) sums up the end behavior of the given polynomial equation.

What is the end behavior of the graph of the polynomial function f(x) = -x⁵ + 9x⁴ - 18x³?

Summary:

It should be noted from the graph above, that between the two endpoints i.e. (y = f(x), - 6, 23328) and (y = f(x), 10, -28000) the curve does depict a flat behavior along the x-axis, but that should not take away anything from the fact that finally as x → ∞ towards the curve also → -∞ and when finally x → -∞ the curve → ∞.