End Behavior, and the Line a Graph Settles Onto

You'll learnto decide which of the three degree cases you are in before reaching for a rule, to read a horizontal asymptote as a claim about the far ends rather than as a fence the graph may not cross, and to compute the two ends separately whenever an even root or an absolute value is in play.

A horizontal asymptote is a claim about the two far ends of a graph, and it is three different claims depending on which degree wins. The line it names is not a barrier — a curve may cross it once, three times, or forever, and still settle onto it. What goes wrong here is almost never the arithmetic. It is applying a remembered rule to the case it does not cover, or computing one end and quietly assuming the other one matches.

8 STEPS · 6 QUICK CHECKS · THREE CASES, TWO ENDS, AND A LINE THAT MAY BE CROSSED · v1

the target is at the ends of the axis ⇒ compare the degrees, then check both ends

Before you start

What you're looking at

One graph in a window that never moves. The horizontal axis runs from −9 to 9 and stays there for all eight steps — nothing zooms out. Two dots leave the middle and travel outward, one toward each end of the axis.

The question

Where are the outputs heading as the inputs run away in each direction — and does the answer at one end have to match the answer at the other?

Watch for

A dashed emerald line marks the level the outputs are settling onto. The short dashed connector each dot drags is the gap still left between the curve and that level; watch its length, not the dot's position.

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