Question

As x → −[infinity], y → ? As x → [infinity], y → ? Determine the end behavior for y = 8x^4 Determine the end behavior for y = -49 + 5x^4 + 3x Determine the end behavior for y = -x^5 + 5x^4 + 5

Answer 1

Problem 1

The end behavior of y = 8x^4 is:

`\text{As x} \to -\infty, \text{ y } \to \infty\\\text{As x} \to \infty, \text{ y } \to \infty`

In either case, y approaches positive infinity. This end behavior is the same as a parabola that opens upward. This applies to any even degree polynomial.

Informally we can describe the end behavior as: "Both endpoints rise up forever".

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Problem 2

The end behavior of y = -49 + 5x^4 + 3x is the exact same as problem 1. Why? Because the degree here is 4. The degree is the largest exponent.

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Problem 3

For this problem we have the polynomial y = -x^5 + 5x^4 + 5

This time the degree is 5, which is an odd number.

The end behavior would be

`\text{As x} \to -\infty, \text{ y } \to \infty\\\text{As x} \to \infty, \text{ y } \to -\infty`

Informally, we can state the end behavior as "Rises to the left, falls to the right".

The endpoints go in opposite directions whenever the degree of the polynomial is odd. Think of a cubic graph. The "falls to the right" is due to the negative leading coefficient.

I strongly recommend using a TI83, TI84, Desmos, or GeoGebra to graph out each polynomial so you can see what the end behavior is doing.