Victoria H.

asked • 01/08/20# Graphing Polynomial Functions

Describe in words how the end behavior of a graph looks.

## 1 Expert Answer

End behavior of a graph can be analyzed in two easy ways.

If the sketch of a graph is given without the function, we can look at the sketch and report what the y-value would be if x is really really big, and report what the y-value would be when the x-value is really really big but negative.

The other way is done algebraically. If the function is given, we can plug in positive infinity for x, and do some "loose algebra" (only considering if the corresponding y value will approach infinity, negative infinity, or zero) to see what y will become for the right side end-behavior. Then, we can plug in negative infinity for x to see what y will become for the left side end-behavior.

For example:

The graph of the parent function parabola y = x^{2.}

Graphically, it looks like a smiley face. As x goes to positive infinity, y is always increasing to positive infinity.

As x goes to negative infinity, y still increases towards positive infinity.

Algebraically, if we plug in positive infinity for x, we get

y = ∞^{2}

Positive infinity squared is Positive infinity.

If we plug in negative infinity for x, we get

y = (-∞)^{2}

Negative infinity squared is positive infinity,

so as x goes to infinity, so does y. as x goes to negative infinity, y goes to positive infinity.

Hope this helps!

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Mark H.

01/08/20