Functions and graphs: Families of functions
A function with a parameter in it
You can draw several straight lines through the origin. In addition, you can show such lines with different slopes by moving the slider. This slider controls the value of the slope indicated by the symbol \(a\). This gives the graph of a line with the equation \(y=a\cdot x\), which depends on a parameter \(a\).
In terms of functions, there is the graph of the parameterized function \(f_a(x)=a\cdot x\). The parameter \(a\) is included in the naming of the function. By varying the parameter you always get a different function whose graph is a straight line through the origin with slope \(a\).
So this actually concerns a family of functions, even though we often speak of a parameterized function as if it were one function.
Often mathematical problems are formulated in a parameterized form and you look for conditions that the parameter must meet. We give a concrete example below.
The graphs of the quadratic function \(g(x)=x^2\) and the parameterized function \(f_p(x)=2x+p \) are a parabola and a line with slope \(2\) that intersects the vertical axis at the point \((0,p)\).
For some parameter values, the graphs have two points in common.
For other parameter values there are no common points and for exactly one parameter value there is one common point.
This follows from the below elaboration of determining common points. \[\begin{aligned}g(x)&=f_p(x)\\[0.25cm] x^2&=2x+p\\[0.25cm] x^2-2x&=p\\[0.25cm] x^2-2x+1&=p+1\\[0.25cm] (x-1)^2&=p+1\\[0.25cm] x-1&=\pm\sqrt{p+1}\\[0.25cm] x&=1\pm\sqrt{p+1}\end{aligned}\] If a negative number arises under the radical sign of a parameter value, there are no common points. If \(p=-1\), then there is one solution, which is \(x=-1\).
