Vertical asymptote - Rational Functions

Vertical asymptote

Basic Concepts
Factoring trinomials
Factoring difference of squares: $x^2 - y^2$

Related Concepts
Infinite limits - vertical asymptotes

Lessons

Notes:

For a rational function: $f(x) = \frac{numerator}{denominator}$

Provided that the numerator and denominator have no factors in common (if there are, we have "points of discontinuity" as discussed in the previous section), vertical asymptotes can be determined as follows:

$\bullet$ equations of vertical asymptotes: x = zeros of the denominator

$i.e. f(x) = \frac{numerator}{x(x+5)(3x-7)}$ ; vertical asymptotes: $x = 0, x = -5, x = \frac{7}{5}$

2.
Identifying Characteristics of Rational Functions
Without sketching the graph, determine the following features for each rational function:

i) point of discontinuity

ii) vertical asymptote

iii) horizontal asymptote

iv) slant asymptote

Vertical asymptote

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AlgebraFactoring trinomials

AlgebraFactoring difference of squares: x2−y2x^2 - y^2x2−y2

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Intro Lesson

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Intro

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Lessons

Notes:

Intro Lesson

1.

Graphing Rational Functions Sketch each rational function by determining: i) vertical asymptote. ii) horizontal asymptotes

a)

f(x)=52x+10f\left( x \right) = \frac{5}{{2x + 10}}f(x)=2x+105​

b)

g(x)=5x2−13x+6−2x2+3x+2g\left( x \right) = \frac{{5{x^2} - 13x + 6}}{{ - 2{x^2} + 3x + 2}}g(x)=−2x2+3x+25x2−13x+6​

c)

h(x)=x320x−100h\left( x \right) = \frac{{{x^3}}}{{20x - 100}}h(x)=20x−100x3​

2.

Identifying Characteristics of Rational Functions Without sketching the graph, determine the following features for each rational function: i) point of discontinuity ii) vertical asymptote iii) horizontal asymptote iv) slant asymptote

a)

a(x)=x−9x+9a(x) = \frac{x - 9}{x + 9}a(x)=x+9x−9​

b)

b(x)=x2−9x2+9b(x) = \frac{x^{2}-9}{x^{2}+9}b(x)=x2+9x2−9​

c)

c(x)=x2+9x2−9c(x) = \frac{x^{2}+9}{x^{2}-9}c(x)=x2−9x2+9​

d)

d(x)=x+9x2−9d(x) = \frac{x+9}{x^{2}-9}d(x)=x2−9x+9​

e)

e(x)=x+3x2−9e(x) = \frac{x+3}{x^{2}-9}e(x)=x2−9x+3​

f)

f(x)=x2+9x+9f(x) = \frac{x^{2}+9}{x+9}f(x)=x+9x2+9​

g)

g(x)=−x−9−x2−9g(x) = \frac{-x-9}{-x^{2}-9}g(x)=−x2−9−x−9​

h)

h(x)=−x2−9−x2+9h(x) = \frac{-x^{2}-9}{-x^{2}+9}h(x)=−x2+9−x2−9​

i)

i(x)=x2−9x+3i(x) = \frac{x^{2}-9}{x+3}i(x)=x+3x2−9​

j)

j(x)=x3−9x2x2−3xj(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}j(x)=x2−3xx3−9x2​

Vertical asymptote

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Factoring trinomials

Algebra
Factoring difference of squares: $x^2 - y^2$

Factoring difference of squares: $x^2 - y^2$

or

Graphing Rational Functions
Sketch each rational function by determining:

i) vertical asymptote.

ii) horizontal asymptotes

$f\left( x \right) = \frac{5}{{2x + 10}}$

$g\left( x \right) = \frac{{5{x^2} - 13x + 6}}{{ - 2{x^2} + 3x + 2}}$

$h\left( x \right) = \frac{{{x^3}}}{{20x - 100}}$

Identifying Characteristics of Rational Functions
Without sketching the graph, determine the following features for each rational function:

i) point of discontinuity

ii) vertical asymptote

iii) horizontal asymptote

iv) slant asymptote

$a(x) = \frac{x - 9}{x + 9}$

$b(x) = \frac{x^{2}-9}{x^{2}+9}$

$c(x) = \frac{x^{2}+9}{x^{2}-9}$

$d(x) = \frac{x+9}{x^{2}-9}$

$e(x) = \frac{x+3}{x^{2}-9}$

$f(x) = \frac{x^{2}+9}{x+9}$

$g(x) = \frac{-x-9}{-x^{2}-9}$

$h(x) = \frac{-x^{2}-9}{-x^{2}+9}$

$i(x) = \frac{x^{2}-9}{x+3}$

$j(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}$