Sketch of y = (x^2 - x + 1)/(x^2 + x + 1) with key features

Key takeaway: This rational function has a nice structure, and it simplifies the sketch a lot once you compare numerator and denominator. The key is to look at end behavior first, then test for intercepts and any symmetry.

Consider

$$y=\frac{x^2-x+1}{x^2+x+1}.$$

1) Horizontal asymptote

The numerator and denominator have the same degree, so the horizontal asymptote is the ratio of leading coefficients:

$$y=1.$$

2) Intercepts

• $y$-intercept: set $x=0$

$$y=\frac{1}{1}=1,$$

so the graph passes through $(0,1)$.

• $x$-intercepts: solve

$$x^2-x+1=0.$$

Its discriminant is

$$(-1)^2-4(1)(1)=1-4=-3<0,$$

so there are no real $x$-intercepts.

3) Compare with the asymptote

Subtract 1:

$$y-1=\frac{x^2-x+1-(x^2+x+1)}{x^2+x+1} =\frac{-2x}{x^2+x+1}.$$

Since

$$x^2+x+1>0 \quad \text{for all real }x$$

because its discriminant is also negative,

the sign of $y-1$ is the sign of $-x$.

So:

• if $x<0$, then $y>1$
• if $x>0$, then $y<1$
• if $x=0$, then $y=1$

That tells you the curve crosses the horizontal asymptote at $(0,1)$.

4) Shape

The curve stays defined for all real $x$ and never hits the $x$-axis. It lies above $y=1$ for negative $x$ and below $y=1$ for positive $x$, with a crossing at $(0,1)$.

If you want a quick extra check, evaluate a couple of values:

$$y(1)=\frac{1-1+1}{1+1+1}=\frac13,$$ $$y(-1)=\frac{1+1+1}{1-1+1}=3.$$

So the sketch passes through $(-1,3)$ and $(1,\tfrac13)$, approaching $y=1$ on both ends.

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Pitfalls the pros know 👇 A frequent mistake is to assume the horizontal asymptote cannot be crossed. That is false here: the graph does cross $y=1$ at $x=0$. Another slip is forgetting to check whether the denominator can be zero; here it cannot, so there are no vertical asymptotes.

What if the problem changes? If the signs in the denominator changed to $x^2-x+1$, then the curve would still have horizontal asymptote $y=1$, but the comparison with 1 would change. Tiny sign changes in rational functions often alter where the graph lies above or below the asymptote, even when the asymptote itself stays the same.

Tags: horizontal asymptote, rational function, curve sketching