LESSON 2-6 (Part A):
FAMILIES OF FUNCTIONS - TRANSLATIONS
TRANSFORMATION: A transformation of a graph is a "shift" or movement/change of that
graph; transformations include horizontal and vertical translations (shift left/right or up/down),
vertical stretch and compression, and reflections
FAMILY OF FUNCTIONS:
PARENT FUNCTION:
Examples:
Sets of functions in which each function is a transformation of
a special function, called the parent function
Simplest form in a set of functions that form a family. Each function
in the family is a transformation of the parent function.
(1) Linear functions - all graphs of linear functions are lines that are transformation
of the parent linear function y = x or (x) = x.
(2) Quadratic functions - all graphs of quadratic functions, called parabolas, are a
transformation of the parent quadratic function y = x2 or (x) = x2.
LINEAR FUNCTIONS
PARENT FUNCTION: y = x or (x) = x
Graph each linear function. Compare it
to the graph of the parent function, y = x,
to the right.
VERTICAL TRANSLATIONS:
1) y = x + 2
2) y = x
3
PARENT FUNCTION: y = x2 or (x) = x2
QUADRATIC FUNCTIONS
VERTICAL TRANSLATION
3) Complete the table: Graph y = x2, y = x2 + 2
y = x2
x
y = x2 + 2
and
y = x2
2 on the same coordinate plane.
y = x2 - 2
-3
-2
-1
0
1
2
3
HORIZONTAL TRANSLATION
4) Complete the table: Graph y = x2, y = (x + 2)2 and y = (x
x
y = x2
y = (x + 2)2
2)2 on the same coordinate plane.
y = (x - 2)2
-3
-2
-1
0
1
2
3
SUMMARY:
TRANSLATIONS
Vertical Translations:
Horizontal Translations:
Translation up k units, k > 0
y = (x) + k
Translation down k units, k > 0
y = (x)
k
Translation right h units, h > 0
y = (x
h)
Translation left h units, h > 0
y = (x + h)
LESSON 2-6 (Part A):
PRACTICE
TRANSLATIONS
1-6: Describe the transformation of the parent function (x).
1) g(x) = (x) + 3
2) h(x) = (x
2)
4) k(x) = (x + 7)
5) m(x) = (x + 4)
2
3) j(x) = (x)
5
6) p(x) = (x
3) + 2
7-9: How is each function related to the parent linear function y = x ?
7) y = x + 5
8) y = x - 3.5
9) y = x +
10-12: How is each function related to the parent quadratic function y = x2 ?
10) y = (x
3)2
11) y = x2 + 7
12) y = (x + 2)2
13-18: Write an equation for each transformation of y = x2.
13) translation up 4 units
14) translation down 3 units
15) translation to the right 7 units
16) translation to the left 2 units
17) translation up 2 units and to the right 5 units
18) translation down 1 unit and to the left 3 units
3