To determine the end behavior of a function, we substitute high values in for x, and we can even substitute in infinity. However, infinity is not a defined number, but it is a concept, so it will give us an idea about the “tails” of the function.
Let’s use logic here:
We know we have a cubic function with a coefficient of 4, so if we input any positive value in for x, we will get a y value that is a large positive value.
For example, let’s plug 1,000,000 in for x:
f(1,000,000)=4(1,000,000)^3
f(1,000,000)=4e+18
This is an extremely high positive value.
Therefore, because we have a positive coefficient and a power of 3, when we input a positive value in for x, we receive a large positive y-value. This means that as x——> ♾️, y——>♾️
Now, let’s look at negative values:
We have already noted that we have a cubic parent function. Remember that a negative times a negative is a positive times a negative is a negative. This is because if we input a negative value in for x, it will be a large negative value. Even with a coefficient of 4, a positive times a negative is a negative. Let’s see an example:
f(♾️)=4(-♾️)^3
f(♾️)=4(-♾️)(-♾️)(-♾️)
f(-♾️)=4(-♾️)^2(-♾️)
f(-♾️)=4(-♾️)^3
So, we end up with negative infinity times four, which is an even larger negative value.
Therefore, x——>-♾️, y——>-♾️