How to sketch a curve with one horizontal asymptote and two turning points

Key concept: A graph is rarely determined by just one asymptote and two turning points. Those features help, but you still need sign changes, intercepts, and end behavior to see the full picture. The same rough description can lead to more than one shape.

Step by step

No — having one horizontal asymptote and two turning points does not guarantee a unique sketch.

What you know from those features:

• the graph levels off toward the horizontal asymptote as $x\to\pm\infty$ or in one direction, depending on the function
• the curve changes direction twice, so there are two turning points

What you still do not know:

• whether the graph crosses the asymptote
• where the intercepts are
• whether the turning points are above or below the asymptote
• whether the function has symmetry
• whether there are any vertical asymptotes or discontinuities

So there can be several different shapes that satisfy the same broad description.

If you are sketching from a formula, use this checklist:

1. Find the domain.
2. Find intercepts.
3. Find horizontal and vertical asymptotes.
4. Locate turning points or stationary points.
5. Check the sign of the function in each interval.
6. Join the pieces smoothly, respecting asymptotes and end behavior.

That extra information is what makes the sketch specific.

Pitfall alert

Students often treat an asymptote plus turning points like a complete fingerprint. It is not. You can easily draw a curve that matches those features but is still wrong because the intercepts or asymptote crossings are off.

Try different conditions

If the question gives the actual equation, then the sketch can become much more definite. For example, a rational function with a horizontal asymptote and two turning points may still be unique once you know its intercepts and any vertical asymptotes. But if you only know the number of turning points and one asymptote, there is usually more than one possible shape.

Further reading

horizontal asymptote, turning point, curve sketching