Look at the right end behavior of each function )\(as x -> infinitey). what about the function determines whether its graph wil point up or down on the right-hand side?
If the first term is a x^n it dominates the terms in smaller powers of x as x becomes large. Therefore just look at that first term. For example
5 x^15 + 3 x^12 + 6 x^7 etc
just look at
5 x^15
that is big y up as x gets big, so it goes up to the right
if it started with -5 x^15 , it would go down to the right
Even positive
When examining the right end behavior of a function (as x approaches infinity), we are interested in understanding whether the graph of the function will point upwards or downwards on the right-hand side. This behavior is determined by the leading term of the function.
To determine the leading term, we look at the highest power of x in the function's formula. This term dominates the behavior of the function as x approaches infinity. There are three possibilities:
1. If the leading term has an even power (e.g., x^2, x^4, etc.), the graph will point upwards on the right-hand side. This is because as x becomes larger, even powers of x will always yield positive values.
2. If the leading term has an odd power and a positive coefficient (e.g., 3x^3, 5x^5, etc.), the graph will also point upwards on the right-hand side. Similarly, as x becomes larger, odd powers of x with positive coefficients will result in positive values.
3. If the leading term has an odd power and a negative coefficient (e.g., -2x^3, -4x^5, etc.), the graph will point downwards on the right-hand side. In this case, as x becomes larger, odd powers of x with negative coefficients will result in negative values.
By identifying the leading term and considering whether its power is even or odd, along with the sign of its coefficient, we can determine the direction in which the graph of the function points on the right side as x approaches infinity.