Rational functions

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Graphs and asymptotes of rational functions, including the reciprocal function \(f\left(x\right)=\frac{1}{x}\), linear rational functions \(\frac{ax+b}{cx+d}\), forms with oblique asymptotes from polynomial division, and rationals with quadratic denominators.

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The reciprocal function 1/x

SL AA 2.8

The reciprocal function is defined by f(x)=x1.

Notice that f(x) is not defined for x=0. In fact, since x1 gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.

And since for very large x, x1 approaches zero, there is also a horizontal asymptote y=0.

Notice also that x11=x, so f(x)=x1 is self-inverse.

Graphs of linear rational functions

SL AA 2.8

A linear rational function has the form

f(x)=cx+dax+b

When the denominator is zero the graph will have a vertical asymptote:

cx+d=0⇒x=−cd🚫

And as x gets very large, the +b and +d can be ignored:

y=f(x)≈cxax=ca🚫

So there is a horizontal asymptote at y=ca.

Rationals with oblique asymptotes

AHL 2.13

When a rational function is of the form

f(x)=dx+eax2+bx+c

there is a vertical asymptote at

x=−de🚫

By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation

y=dax+C

for some constant C determined during the polynomial division.

Rationals with quadratic denominator

AHL 2.13

When a rational function is of the form

f(x)=cx2+dx+eax+b

There will be vertical asymptotes when the quadratic cx2+dx+e=0.

The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.

Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.

Note that if the numerator and denominator share a root (ie x=−ab is a root of the denominator), then there will only be one asymptote and a "hole" on the x-axis. This has never shown up on exams.

Nice work completing Rational functions, here's a quick recap of what we covered:

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Mixed Practice

Exercises checked off

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Mixed Practice

Transformations & asymptotes

Rational functions

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Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

The reciprocal function 1/x

SL AA 2.8

The reciprocal function is defined by f(x)=x1.

Notice that f(x) is not defined for x=0. In fact, since x1 gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.

And since for very large x, x1 approaches zero, there is also a horizontal asymptote y=0.

Notice also that x11=x, so f(x)=x1 is self-inverse.

Graphs of linear rational functions

SL AA 2.8

A linear rational function has the form

f(x)=cx+dax+b

When the denominator is zero the graph will have a vertical asymptote:

cx+d=0⇒x=−cd🚫

And as x gets very large, the +b and +d can be ignored:

y=f(x)≈cxax=ca🚫

So there is a horizontal asymptote at y=ca.

Rationals with oblique asymptotes

AHL 2.13

When a rational function is of the form

f(x)=dx+eax2+bx+c

there is a vertical asymptote at

x=−de🚫

By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation

y=dax+C

for some constant C determined during the polynomial division.

Rationals with quadratic denominator

AHL 2.13

When a rational function is of the form

f(x)=cx2+dx+eax+b

There will be vertical asymptotes when the quadratic cx2+dx+e=0.

The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.

Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.

Nice work completing Rational functions, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Mixed Practice

Transformations & asymptotes

Rational functions

0 of 0 exercises completed

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

The reciprocal function 1/x

SL AA 2.8

The reciprocal function is defined by f(x)=x1.

Notice that f(x) is not defined for x=0. In fact, since x1 gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.

And since for very large x, x1 approaches zero, there is also a horizontal asymptote y=0.

Notice also that x11=x, so f(x)=x1 is self-inverse.

Graphs of linear rational functions

SL AA 2.8

A linear rational function has the form

f(x)=cx+dax+b

When the denominator is zero the graph will have a vertical asymptote:

cx+d=0⇒x=−cd🚫

And as x gets very large, the +b and +d can be ignored:

y=f(x)≈cxax=ca🚫

So there is a horizontal asymptote at y=ca.

Rationals with oblique asymptotes

AHL 2.13

When a rational function is of the form

f(x)=dx+eax2+bx+c

there is a vertical asymptote at

x=−de🚫

By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation

y=dax+C

for some constant C determined during the polynomial division.

Rationals with quadratic denominator

AHL 2.13

When a rational function is of the form

f(x)=cx2+dx+eax+b

There will be vertical asymptotes when the quadratic cx2+dx+e=0.

The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.

Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.

Nice work completing Rational functions, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

1 free

Mixed Practice

Transformations & asymptotes

Rational functions

0 of 0 exercises completed

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

The reciprocal function 1/x

SL AA 2.8

The reciprocal function is defined by f(x)=x1.

Notice that f(x) is not defined for x=0. In fact, since x1 gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.

And since for very large x, x1 approaches zero, there is also a horizontal asymptote y=0.

Notice also that x11=x, so f(x)=x1 is self-inverse.

Graphs of linear rational functions

SL AA 2.8

A linear rational function has the form

f(x)=cx+dax+b

When the denominator is zero the graph will have a vertical asymptote:

cx+d=0⇒x=−cd🚫

And as x gets very large, the +b and +d can be ignored:

y=f(x)≈cxax=ca🚫

So there is a horizontal asymptote at y=ca.

Rationals with oblique asymptotes

AHL 2.13

When a rational function is of the form

f(x)=dx+eax2+bx+c

there is a vertical asymptote at

x=−de🚫

By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation

y=dax+C

for some constant C determined during the polynomial division.

Rationals with quadratic denominator

AHL 2.13

When a rational function is of the form

f(x)=cx2+dx+eax+b

There will be vertical asymptotes when the quadratic cx2+dx+e=0.

The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.

Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.

Nice work completing Rational functions, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

1 free