Parent and Family Graphs
Objective
Use a graphing calculator to see how the graphs of quadratic
functions change as the coefficients change.
In this lesson, we will start with the parent function y = x^{
2} and study how the graphs change in shape or in position
when the coefficients are changed. You should use graphing
calculators or computers to graph the families of functions, and
try to find similarities and differences between them.
Multiplying x^{2} by a Positive Number
Graph y = x^{ 2} and then the following family of
functions.
Notice how the sizes of these parabolas change as the
coefficient of x^{ 2} increases or decreases. Make sure
you understand the following characteristics.
1. The vertices and the axes of symmetry of all these
parabolas are the same. Namely, the axis of symmetry is the line
x = 0, and the coordinates of the vertex are (0, 0). This is
because in a quadratic function of the form y = ax^{ 2} +
0 x + 0, b = 0, so the axis of symmetry is given by or 0, which is the y axis.
2. The graphs of the quadratic functions y = ax^{ 2} ,
where a > 0, get more narrow as the value of a increases. The
graphs get wider as the value of a decreases.
Why do parabolas change sizes in this way? The larger the
value of a in y = ax^{ 2} , the faster the y variable
increases as x grows. To see this, make a table of values of the
functions , y = x^{ 2} ,
and y = 5 x^{ 2} . It is clear from this table that the
larger the coefficient of x^{ 2} , the faster the y value
grows. This results in a steeper incline of the graph, or a
“narrower” parabola. Similarly, the smaller the
coefficient of x^{ 2} , the more slowly the y value
grows. This results in a more gradual incline of the graph, or a
“wider” parabola.
What happens as the coefficient a in y = ax^{ 2} gets
smaller and smaller (but remains positive), until it approaches
0? As a approaches 0, the parabolas get wider and wider until
they approach a flat line, namely the graph of y = 0x or 0, which
is the xaxis.
Multiplying x^{ 2} by a Negative Number
Now what happens to the graphs of y = ax^{ 2} when a
is negative? Graph the following family of functions.
When the a is negative in y = ax^{ 2} , the parabolas
open downwards. This is because as x grows, the y variable
becomes more and more negative. So the graphs incline downward.
Notice that these graphs have the same characteristics that we
saw before.
1. These parabolas all have the same axes of symmetry and
vertices. The axis of symmetry is still the vertical line x = 0
(or the y axis), and the vertices are still at (0, 0). This is
because in these functions y =  ax^{ 2} + 0x + 0, or 0. Notice however that the vertices
are now maximum points of the graphs, where in the earlier
examples they were minimum points.
2. Similar to previous examples, as the coefficient a in y =
ax^{ 2} becomes smaller (more negative), the parabola
becomes more narrow. As a gets closer to zero, the parabola gets
wider. This is because the smaller the coefficient a is, the
faster the y value decreases (gets more negative) as the x value
grows. Have your students make tables of values of the functions
graphed above to verify this.
