Rational Functions

Long Division of Polynomials #

The same four moves, with x in place of digits.

Dividing one polynomial by another uses the same divide, multiply, subtract steps as long division of numbers

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Long division asks how many times x + 2 goes into $x^2+5x+7$x² + 5x + 7.

Compare the leading terms only. $x^2/x$x² / x is x, so the answer starts with x.

Multiply back and subtract: the $x^2$x² cancels, leaving 3x + 7.

Repeat on what is left: $3x/x=3$3x / x = 3, and the remainder is 1.

The quotient is x + 3, and the remainder 1 is written over the divisor.

Zeros and Vertical Asymptotes #

The top decides one, the bottom decides the other.

A rational function is zero where its numerator is zero and has a vertical asymptote where its denominator is zero

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A rational function is one polynomial written over another.

A fraction is zero only when its numerator is zero, so the graph crosses at x = 1.

At x = 3 the denominator is zero, and division by zero is undefined.

Just above 3 the denominator is tiny and positive, so the values climb without bound.

In symbols: to the right of 3 the curve rises without bound, and to the left it falls without bound.

Holes Where a Factor Cancels #

One point missing from an otherwise ordinary curve.

A factor shared by the top and the bottom leaves a single missing point rather than an asymptote

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Factor the numerator and the denominator first. The same bracket appears in both.

Cancel and x + 2 is left — but at x = 2 the original reads 0 divided by 0.

So the graph is the line y = x + 2 with exactly one point missing: a hole.

Change the denominator and the shared bracket is gone, so nothing cancels.

Now the denominator is 0 at x = 3 with nothing to cancel it, so the curve has a vertical asymptote.

Horizontal Asymptotes by Degree #

Far out, only the leading terms are left.

Far from the origin only the leading terms matter, so the two degrees decide the horizontal asymptote

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Divide numerator and denominator by x, and the rest becomes fractions in $\frac{1}{x}$1/x.

As x grows, $\frac{1}{x}$1/x and $\frac{3}{x}$3/x tend to 0, leaving 2 ÷ 1 = 2.

So the curve approaches y = 2, from below on one side and from above on the other.

When the denominator has the higher degree, the fraction tends to 0, so the asymptote is y = 0.

Three cases, and comparing the two degrees is what picks between them.

Slant Asymptotes by Long Division #

Divide, and a line falls out of the quotient.

When the numerator’s degree is one more than the denominator’s, long division gives a line plus a remainder that tends to 0

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The numerator’s degree is one more than the denominator’s, so the values grow and no horizontal line fits.

Long division, exactly as before: the quotient starts with x, and 2x + 5 is left.

One more step gives 2, with a remainder of 3.

As x grows, the remainder term $\frac{3}{x+1}$3/(x + 1) tends to 0, leaving the line y = x + 2.

The curve approaches that slant line on both sides and never meets it.

Values a Rational Function Cannot Take #

The discriminant marks the heights a curve never reaches.

Rearrange y = f(x) into a quadratic in x: y is reached only when that quadratic has a real root

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$y=\frac{x^2+1}{x}$y = (x² + 1)/x never takes a value between −2 and 2: no point of the curve lies in that band.

To find the heights it can reach, multiply by x and rearrange into a quadratic in x.

A height y is reached only if that quadratic has a real root, so its discriminant is at least 0.

At the boundary the discriminant is 0, so there is one x: the edges of the band are the turning points.