Pre-Calculus
TRANSFORMATIONS EXPLORATION
You will need your graphing calculator, graph paper, a ruler and colored pencils for this exploration. In this lesson, you will
learn about PARENT FUNCTIONS and how changing a function rule will alter the graph of a function through
TRANSFORMATIONS.
Vocabulary: A PARENT FUNCTION is the most basic function of a family of functions, or the original function before a
transformation is applied. For example: y = x2 is a parent function. The graph of the function y = x2 - 3 is a
transformation of the parent function, or an alteration of the graph.
1) Carefully draw A COORDINATE PLANE on your paper that goes from –4 to 4 on the x- and y-axis. You will use this
COORDINATE PLANE for problems a - f:
a)
b)
c)
d)
e)
f)
Using your calculator, graph y = x. This is called the LINEAR FUNCTION.
Carefully draw and label this graph on your coordinate plane. Use a ruler!
Using your calculator, graph the function y = x – 2.
Draw this graph on your coordinate plane with a colored pencil.
On the same coordinate plane, draw and label y = x + 3 with a different colored pencil.
The transformations performed on the function y = x are called TRANSLATIONS or VERTICAL SHIFTS. Below
your coordinate plane, write “Vertical shift down 2” and “Vertical shift up 3”.
2) Carefully draw A COORDINATE PLANE on your paper that goes from –4 to 4 on the x- and y-axis. You will use this
COORDINATE PLANE for problems a - f:
a)
b)
c)
d)
e)
f)
Using your calculator, graph y = x2. This is called the QUADRATIC FUNCTION.
Carefully draw and label this graph on your coordinate plane.
Using your calculator, graph the function y = (x + 2)2.
Draw this graph on your coordinate plane with a colored pencil.
On the same coordinate plane, draw and label y = (x - 3)2 with a different colored pencil.
The transformations performed on the function y = x2 are called TRANSLATIONS or HORIZONTAL SHIFTS.
Below your coordinate plane, write “Horizontal shift left 2” and “Horizontal shift right 3”.
3) Carefully draw A COORDINATE PLANE on your paper that goes from –4 to 4 on the x- and y-axis. You will use this
COORDINATE PLANE for problems a - f:
a)
b)
Using your calculator, graph y = x . This is called the RADICAL (SQUARE ROOT) FUNCTION.
Carefully draw and label this graph on your coordinate plane.
c)
d)
Using your calculator, graph the function y = - x .
Draw this on your coordinate plane with a colored pencil.
e)
On the same coordinate plane, draw and label y =
f)
The transformations applied to y = x are called REFLECTIONS. Below your coordinate plane, write

“Reflection across the x-axis” and “Reflection
across the y-axis”.
x
with a different colored pencil.

4) Carefully draw A COORDINATE PLANE on your paper that goes from –4 to 4 on the x- and y-axis. You will use this
COORDINATE PLANE for problems a - f:
a) Using your calculator, graph y = | x | in Y1. This is called the ABSOLUTE VALUE FUNCTION.
Hint: go to MATH  NUM 1: abs(
b) Carefully draw and label this graph on your coordinate plane. Use a ruler!
c) Using your calculator, graph the function y = 2| x |. Draw this on your coordinate plane with a colored pencil.
d) This is called a VERTICAL STRETCH BY A FACTOR OF 2. Label this transformation on your graph.
e) Using your calculator, graph the function y =(1/2) | x |. Draw this on your coordinate plane with a different
colored pencil.
f)
This is called a VERTICAL COMPRESSION BY A FACTOR OF
1
. Label this transformation on your graph.
2
5) Carefully draw A COORDINATE PLANE on your engineering paper that goes from –4 to 4 on the x- and y-axis. You
will use this COORDINATE PLANE for problems 25 - 28:
a) Using your calculator, graph y = x3 in Y1. This is called the CUBIC FUNCTION.
b) Carefully draw and label this graph on your coordinate plane. Use a ruler!
3
c) Using your calculator, graph y = (2x) .
d) Carefully draw and label this graph on your coordinate plane.
e) This is called a HORIZONTAL COMPRESSION BY A FACTOR OF
Label this transformation
 on your graph.
f)
Using your calculator, graph y =
1
. Be careful! The factor is not 2, it is ½.
2
1
( x) 3 .
2
g) Carefully draw and label this graph on your coordinate plane.
h) This is called a HORIZONTAL STRETCH BY A FACTOR OF 2. Be careful! The factor is not 1/2, it is 2. Label this
transformation on your graph.

6) Carefully draw A COORDINATE PLANE on your paper that goes from –4 to 4 on the x- and y-axis. You will use this
COORDINATE PLANE for problems a - d:
a) Using your calculator, graph y =
1
. This is called the RECIPROCAL (RATIONAL) FUNCTION.
x
b) Carefully draw and label this graph on your coordinate plane. Use a ruler!
c)
Think about it! What will happen if you graph y =
1
 2 ? Graph, using a colored pencil, what you think
( x  3)
this function will look like. Then, check it with your calculator.
d) Label the combination of transformations that were applied.
7) ON A SEPARATE PAPER: For each of the following, identify the parent function and describe the transformations that
have occurred to the parent function.
a) y =
2(x 1) 2
b) y =
 ( x  4)3
c)
y=
x 3
d) y =
1
2
(x 1)
e) y =
x 3 2

f)
y=

1
 x
2
g) y =
 2x
h) y =
2
3
x
i)
y=
(x  3)2  4
j)
y=
x 5 3
k)
y=
(x  2) 3 1
l)
y=
 x3 2
