definition
A parent function is the simplest function of a family of functions.
For the family of quadratic functions,  y = ax2 + bx + c, the simplest function
of this form is y = x2.

button The "Parent" Graph: parentgraph
The simplest parabola is y = x2, whose graph is shown at the right. The graph passes through the origin (0,0), and is contained in Quadrants I and II.

This graph is known as the "Parent Function" for parabolas, or quadratic functions. All other parabolas, or quadratic functions, can be obtained from this graph by one or more transformations.

button The "Children" Graphs:
 quadfamilypic
The "parent" parabola can give birth to a myriad of other parabolic shapes through the process of transformations.

Brush off your memories of transformations and let's take a quick look at what is possible.

When graphing quadratic functions (parabolas), keep in mind that two forms of equations may be used:
    y = ax2 + bx + c   or   y = a(x - h)2 + k

 
Vertical Translation
Horizontal Translation
move the graph vertically - up or down
y = x2+ k
y = x2+ 4 moves the graph UP 4 units
y = x2- 4 moves the graph DOWN 4 units

transgraph1
move the graph horizontally - left or right
y = (x - h)2
y = (x - 4)2 moves the graph RIGHT 4 units
y = (x + 2)2 moves the graph LEFT 2 units

transgraph2
Reflection in x-axis
Reflection in y-axis
flip the graph over the x-axis
(negates the y-values of the coordinates)
y = -(x2)
y = x2 parent graph
y = -(x2) parent reflected over x-axis

transgraph3

flip the graph over the y-axis
(negates the x-values of the coordinates)
y = (-x)2
y = x2 parent graph
y = (-x)2 parent reflected over y-axis

transgraph4
Stretch or Compress Vertically
Stretch or Compress Horizontally
stretches away from the x-axis or compresses toward the x-axis
y = a • x2
| a | > 1 is a stretch;
0 < | a | <1 is a compression

y = x2 parent graph
y = ¼(x)2 vertical compression
y = 4(x)2 vertical stretch
transgraph5a
stretches away from the y-axis or compresses toward the y-axis
y = (a • x)2
| a | > 1 is a compression by factor of 1/a;
0 < | a | <1 is a stretch by factor of 1/a

y = x2 parent graph
y = (¼x)2 horizontal stretch
y = (4x)2 horizontal compression
transgraph6




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