6.1.1 An introductory example
Every function we have studied so far has had one input and one output. Usually our variable for the input is and our variable for the output is . To indicate that our output variable is dependent on our choice of input variable we write , literally read " is a function of ".
In this unit we will introduce a new type of function, one that has one input, usually named and two outputs, usually named and . Our two output variables are dependent on our choice of input variable. We will often write and indicating that and are each functions of . Later when you take my Multivariable Calculus class I will teach you to write indicating that the function inputs a single number and outputs a two-dimensional point .
We refer to the functions and as parametric equations, and the variable is called the parameter. In the GeoGebra file below you can practice plotting the output of such a function by plugging in various values for the parameter.
What point is plotted when ?
What do the arrows represent?
The resulting picture is the graph of a function of the form . What is the equation and the domain of that function? [Hint: Start with the equation for and rearrange to have an equation for . Then perform a substitution in the equation for to obtain an equation relating only and ).
The parametric equations and ended up plotting the graph of a line segment. Describe how to find the slope of this line segment?