02.20othergraphs

Graphs of Other Algebraic Functions

Rational Functions

For the purpose of this course, a rational function is:

… … any function in the form: $y = \dfrac{ f(x) }{ g(x) }$

… … where f(x) and g(x) are polynomials

The rational function above will have a maximal or implied domain of R excluding the values where $g(x) = 0$

The simplest rational function is the rectangular hyperbola with vertical and horizontal asymptotes

… … The Standard Hyperbola: $y = \dfrac{1}{x}$ Another common rational function is the truncus with vertical and horizontal asymptotes

The positive truncus looks like the trunk of a tree.

… … The Standard Truncus: $y = \dfrac{1}{x^2}$ Square Root and Cube Root Functions

… … The Standard Square Root Function: $y = x^\dfrac{1}{2} \qquad \text{or } y = \sqrt{x}$ … … The Standard Cube Root Function: $y = x^\dfrac{1}{3} \qquad \text{or } y = \sqrt{x}$ Graphs of y = xn where n is rational (a fraction)

The square root and cube root graphs are special cases of the group of graphs with a rational exponent.

If the exponent is the fraction $\dfrac{p}{q}$ then the shape of the graph will depend on

• whether p and q are odd or even
• whether p/q < 1 or p/q > 1

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