Parent and Family Graphs
Adding a Constant
Now consider a family of quadratic functions that are found by
adding a constant to y = x^{ 2} . Graph the following
functions on graphing calculators.
y = x^{ 2}, y = x^{ 2}  2, y = x ^{2}
+ 2, y = x^{ 2} + 4
Try to find out what happens to the graphs of quadratic
functions of the form y = x^{ 2} + c as c changes. Make
sure your students notice the following.
1. The axis of symmetry for each parabola is the same, namely,
the vertical line x = 0, which is the y axis. This is because in
y = x^{ 2} + c = 1x^{ 2} + 0x + c , a = 1, and b
= 0. So, the axis of symmetry is the vertical line or 0.
2. The vertex of the parabola moves up or down, depending on
the value of c in y = x^{ 2} + c . Since the axis of
symmetry is the line x = 0 (or the y axis), the x coordinate of
the vertex is 0. To find the y coordinate, substitute x = 0 into
the equation.
y = x^{ 2} + c
y = 0^{ 2 }+ c Replace x with 0.
y = c
So the y coordinate of the vertex is c . In other words, the
vertex of the parabola given by the function y = x^{ 2} +
c is the point at (0, c ). So if we add c to y = x^{ 2} ,
the vertex moves up by c . If we subtract c from y = x^{ 2}
, the vertex moves down by c .
3. The parabolas all have the same size. They have just been
shifted up or down by c units because the y values of the
function y = x^{ 2} + c are exactly c units more than the
y values of the function y = x^{ 2} . This is called a vertical
translation. You can see this pattern with a table like
the one shown below.
Adding a Constant Before Squaring
Now consider the family of quadratic functions of the form y =
( x + d )^{ 2} , where d is some constant. Graph the
following examples on graphing calculators.
y = ( x  2)^{ 2} , y = ( x  1)^{ 2} , y = x^{
2} , y = ( x + 2)^{ 2}
Try to figure out what happens to the graphs of quadratic
functions of the form y = (x + d)^{ 2} as d changes. Make
sure you understand the following.
1. The axis of symmetry shifts to the left or right, depending
on whether d is positive or negative. To understand this
algebraically, expand the expression y = ( x + d )^{ 2} =
x^{ 2} + 2dx + d^{ 2} . In this expression, a =
1, b = 2d , and c = d^{ 2}. So the axis of symmetry has
the equation . For example, if d is
positive, the line x =  d is the vertical line shifted to the
left d units from the y axis.
2. The vertex of the parabola moves left or right, depending
on the value of d in y = ( x + d )^{ 2} . Since the axis
of symmetry is the line x =  d , the xcoordinate of the vertex
is  d . To find the y coordinate, substitute x = d into the
equation.
y = ( x + d )^{ 2}
= (  d + d )^{ 2} Replace x with  d .
= 0^{ 2} or 0
So the ycoordinate of the vertex is 0. In other words, the
vertex of the parabola given by the function y = ( x + d )^{ 2}
is the point at (  d , 0).
3. The parabolas all have the same size. They have just been
shifted to the right or left by d units. This is called a
horizontal translation .
