Graphing Technology: Parent and Family Graphs
Objective Use a graphing calculator to
explore how changing the values of m and b affect the graph of y
= mx + b .
In this lesson, you will be asked to explore the graphs of
linear equations, and how they depend on the slope and the
y-intercept. There are several main ideas that this lesson
Lines with positive slope have the property that as we
move to the right along the line, the line slopes upward.
Lines with negative slope slope downward.
Lines with zero slope are horizontal.
Lines with greater positive slope move upward more
steeply. Lines with lesser negative slope move downward more
Try to explore both positive and negative values of m by
graphing equations of the form y = mx . The collection of lines
obtained in this way can be thought of as a family of lines. A
family of lines is shown below.
Any one of the lines in the family can be thought of as a parent,
since the other members of the family are obtained by rotating
that line. Typically a simple member, such as y = x , is the
In graphing an equation of the form y = mx + b ,
the line shifts up as the value of b increases, and
the line shifts down as the value of b decreases.
Try to explore by graphing equations of the form y = x + b for
various values of b, both positive and negative. The lines are
simply a family of parallel lines that move up or down. Again, a
simple member, such as y = x , is the parent graph. A family of
graphs is shown below.
Next, try to explore the following different families of
lines, y = 2x + b and y = -x + b , for various values of b. You
will again find families of parallel lines, with slopes 2 and -1,
What can you say about lines having the same
The answer to this question is that they are parallel.