What is the end behavior of the graph of the polynomial function f(x) = 2x³ - 26x - 24?

Solution:

The end behaviour of a polynomial function implies that how f(x) behaves when x tends to 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) = 2x³ - 26x - 24 is given below:

If we look at the upper end of the graph i.e. (y = f(x), 6, 252), it can be seen that it is moving upwards(north direction). As the value of ‘x’ increases towards the right 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), -6, -300) , it can be seen that it is moving downwards (south 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 behaviour of the given polynomial equation.

What is the end behavior of the graph of the polynomial function f(x) = 2x³ - 26x - 24?

Summary:

It should be noted from the graph above, that between the two end points i.e. (y = f(x), 6, 252) and (y = f(x), -6, -300) the curve does depict a periodical behaviour on either side of the y-axis, but that should not take away anything from the fact that finally as x → ∞ the curve also → ∞ and when x → -∞ the curve also → -∞.