Rational Functions

Stage 10 of 23 Strand 2 of 9 5 lessons

5 illustrated lessons, each teaching the why before the how.

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Long Division of Polynomials

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

Dividing one polynomial by another runs the same divide, multiply, subtract loop as numbers

Ask what long division always asks: how many of x + 2 fit into this?

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

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

Run the loop again on what is left: 3 more, and a remainder of 1.

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

Now you

Divide x² + 4x + 5 by x + 3. What is the quotient?

Divide x² + 7x + 15 by x + 3. What is the remainder?

Zeros and Vertical Asymptotes

The top decides one, the bottom decides the other.

A rational function is zero where its top is zero and runs away where its bottom is zero

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: the curve runs up on the right of 3 and down on the left.

Now you

Where does y = (x + 2)/(x − 5) cross the x-axis?

Where does y = (x − 1)/(x − 5) cross the x-axis?

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

Factor both parts first. The same bracket appears above and below.

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 really does vanish, and the curve runs away instead.

Now you

y = (x − 1)(x + 2)/(x − 1). What sits at x = 1?

y = (x − 5)(x + 6)/(x − 5). What sits at x = 5?

Horizontal Asymptotes by Degree

Far out, only the leading terms are left.

Far from the origin the leading terms decide everything, so the two degrees fix the flat line

Divide numerator and denominator by x, and the rest becomes fractions in 1/x.

Those fractions shrink to nothing as x grows, leaving 2 over 1.

So the curve flattens onto y = 2 — from below one way, from above the other.

With a bigger bottom the fraction fades to nothing, so the flat line is y = 0.

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

Now you

What is the horizontal asymptote of y = (x + 4)/(x² + 1)?

What is the horizontal asymptote of y = (x + 3)/(x² + 5)?

Slant Asymptotes by Long Division

Divide, and a line falls out of the quotient.

When the top is one degree bigger, long division splits the function into a line and a vanishing remainder

The top is one degree bigger, 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 turn of the loop gives 2, with a remainder of 3.

Far out the remainder piece fades to nothing, leaving the line y = x + 2.

The curve closes onto that slant line from both sides and never meets it.

Now you

What is the slant asymptote of y = (x² + 4x + 8)/(x + 3)?

Why does y = (x² + 3x + 6)/(x + 1) have no horizontal asymptote?

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