How to Graph the Reciprocal of a Quadratic Function

Take a quadratic function \(f(x) = ax^2+bx+c\) and flip it upside down to form its reciprocal, \(g(x) = \frac{1}{f(x)}\). The result is a completely different-looking graph made of smooth curved branches instead of a single parabola. The idea builds on the same logic you used when graphing reciprocals of linear functions, except now the denominator can have zero, one, or two roots, which changes how many vertical asymptotes the graph has.

Before plotting a single point, gather three pieces of information about the original quadratic \(f(x)\).

• Zeros of \(f(x)\): every value of \(x\) where \(f(x)=0\) becomes a vertical asymptote of \(g(x)\), since dividing by zero is undefined.
• Vertex of \(f(x)\): the y-value of the vertex reciprocates. A minimum of \(f(x)\) becomes a maximum of \(g(x)\), and a maximum of \(f(x)\) becomes a minimum of \(g(x)\).
• Sign of \(f(x)\) in each interval: wherever \(f(x)\) is positive, \(g(x)\) is positive too, and wherever \(f(x)\) is negative, \(g(x)\) is negative.

No matter which quadratic you start with, \(g(x)\) always has a horizontal asymptote at \(y=0\), because as \(x\) moves far from the origin \(f(x)\) grows without bound, and \(\frac{1}{f(x)}\) shrinks toward zero.

The overall shape of \(g(x)\) depends entirely on how many real zeros \(f(x)\) has.

• Two distinct real roots: \(g(x)\) has two vertical asymptotes and three separate branches.
• One repeated root: \(g(x)\) has a single vertical asymptote, and both sides of it shoot off in the same direction because the quadratic touches zero without changing sign.
• No real roots: \(f(x)\) never equals zero, so \(g(x)\) has no vertical asymptote at all and forms one smooth, continuous bump or dip.

Graph \(g(x) = \dfrac{1}{x^2-4}\).

Step 1. Factor the denominator: \(x^2-4=(x-2)(x+2)\), so \(f(x)=0\) at \(x=-2\) and \(x=2\). These are the vertical asymptotes.

Step 2. The vertex of \(f(x)=x^2-4\) is at \((0,-4)\). Since \(f(0)=-4\), the reciprocal passes through \(\left(0,-\frac14\right)\).

Step 3. Check the sign of \(f(x)\) in each interval created by the roots.

Step 4. Between the asymptotes, \(f(x)\) is negative, so \(g(x)\) stays negative, reaching its highest point (closest to zero) at \(\left(0,-\frac14\right)\) and diving toward negative infinity near \(x=-2\) and \(x=2\). Outside the asymptotes, \(f(x)\) is positive and growing, so \(g(x)\) stays positive and shrinks toward zero as \(x\) moves further out.

Graph \(g(x) = \dfrac{1}{x^2+1}\).

Since \(x^2 \ge 0\) for every real \(x\), \(f(x)=x^2+1\) is always at least \(1\), so it never equals zero. That means \(g(x)\) has no vertical asymptote anywhere, and it stays positive for all \(x\).

The vertex of \(f(x)\) is at \((0,1)\), the smallest value \(f(x)\) can take. Since the smallest denominator gives the largest fraction, the reciprocal reaches its overall maximum right there, at \((0,1)\). As \(x\) moves away from zero in either direction, \(f(x)\) grows, so \(g(x)\) sinks smoothly toward the horizontal asymptote \(y=0\) without ever reaching it.

Graph \(g(x) = \dfrac{1}{(x-1)^2}\).

Here \(f(x)=(x-1)^2\) has a repeated root at \(x=1\), so there is exactly one vertical asymptote there. Because \((x-1)^2\) is a square, it is always zero or positive, and it never turns negative on either side of \(x=1\). That means \(g(x)\) is positive everywhere it is defined, and both branches rise toward positive infinity as \(x\) approaches \(1\) from the left and from the right, rather than one branch going up and the other going down.

Far from the asymptote, \(f(x)\) grows quickly, so \(g(x)\) again settles toward the horizontal asymptote \(y=0\).

Whenever you need to graph the reciprocal of a quadratic function, work through the same checklist every time: factor to find the zeros of the quadratic and draw the vertical asymptotes, locate the vertex and reciprocate its y-value to place a turning point, run a quick sign check across each interval, and remember that \(y=0\) is always the horizontal asymptote. Once these pieces are marked, connecting them with smooth curved branches gives an accurate sketch without plotting dozens of individual points.

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