linear functions - parameter plane
A linear function in the x,y plane can be written as y = mx + b.
For example, the linear function y = 2x +3, has the values m = 2 and b = 3.
We can plot this point (2,3) in the m,b plane.
Every linear function in the x,y plane corresponds to a point in the m,b plane.
Exploration: check the EXPLORE box and experiment to see how this works.
What points in the m,b plane correspond to the functions
y = b, b = -1, 0, 1, 2, 3, ... in the x,y plane?
What points in the m,b plane correspond to the functions
y = mx, m = -1, 0, 1, 2, 3, ... in the x,y plane?
Challenge: check the CHALLENGE box - a dotted line will appear in the m,b plane.
You can position this dotted line by dragging the two small black rings.
After you have placed the dotted line where you want it,
you can slide a large RED dot along the dotted line.
Each position of the large RED dot has a set of coordinates (m,b)
and therefore corresponds to a linear function in the x,y plane.
As you slide the RED dot along the dotted line you will notice that
all the linear functions in the x,y plane pass through a single point.
YOUR CHALLENGE: How is the position of the fixed point in the x,y plane
related to the way you placed the dotted line in the m,b plane?