Date:________________________
Block:__________ Name:____________________________________
The Properties of Graphs of Exponential Functions
Part A- Reflections in the x-axis
 Graph the exponential y  2 x by completing the
following chart:
x
y
10
y
9
8
-3
7
-2
6
-1
5
0
4
1
3
2
2
1
3
x
-4

-3
-2
-1
1
2
3
4
1. What is the y value where the graph seems to level off?____________________
This line is called the horizontal asymptote.
2. Does the graph produce a growth curve or a decay curve?__________________
3. What is the focal point of this curve?_____________________

x
Graph the following function, y  (2 x )
y
x
-4
-3
-2
-1
1
2
3
4
-1
-3
-2
-2
-3
-1
-4
0
-5
1
-6
2
-7
3
-8


Does the graph produce a growth curve or a decay
curve?_________________
What is the focal point of this curve?_____________
-9
-10
y
4. Make a statement connecting the sign of the coefficient of 2 x and the look of the graph.
*This graph is said to be a reflection in the x-axis of the y  2 x graph.
1
Date:________________________ Block:__________ Name:____________________________________
Part B- Reflections in the y-axis
 Graph the exponential y  2 x by completing the
10 y
following chart:
9
x
y
8
-3
7
-2
6
-1
5
0
4
1
3
2
2
3
1
x
-4
-3
-2
-1
5. Does the graph produce a growth curve or a
decay curve?__________________
6. What is the focal point of this curve?_____________________

x
Graph the following function, y  2 x
y
1
10
2
3
4
y
9
-3
8
-2
7
-1
6
0
5
1
4
2
3
3
2
1
x


Does the graph produce a growth curve or a decay -4
curve?_________________
What is the focal point of this curve?_____________
-3
-2
-1
1
2
3
4
7. Make a statement connecting the sign of the coefficient in front of the x term of 2 x and the look
of the graph.
*This graph is said to be a reflection in the y-axis of the y  2 x graph.
2
Date:________________________ Block:__________ Name:____________________________________
Part C- Vertical Stretches
 Complete the table of values for y  2 x and
10 y
1
y  2 x below, and draw both graphs on the right.
3
9
x
1
y2
y  2x
8
3
x
y
x
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
7
y
6
5
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
1. How are the two graphs alike? (shape, growth/decay, steepness, etc.)
______________________________________________________________________________
______________________________________________________________________________
2. How are they different? (focal point, etc.)
______________________________________________________________________________

Complete the table of values for 3 y  2 x and
y  2 x below, and draw both graphs on the right.
10
y
9
y2
x
x
y
3y  2
x
x
y
8
7
-3
-3
6
-2
-2
5
-1
-1
4
0
0
3
1
1
2
2
2
1
3
3
x
-4
-3
-2
-1
1
2
 This factor of change between the y-values of
1
y  2 x and y  2 x is known as a vertical stretch. In this case, we have a vertical stretch of 3.
3
1
 3 y  2 x has a vertical stretch of from y  2 x
3
3
3
4
Date:________________________ Block:__________ Name:____________________________________
Part D- Horizontal Stretches
 Complete the table of values for y  2 x and
10 y
y  2( x / 2) below, and draw both graphs on the right.
y  2x
y  2( x / 2)
9
x
x
8
y
y
-3
-3
7
-2
-2
6
-1
-1
5
0
0
4
1
1
3
2
2
2
3
3
1
x
-4
-3
-2
-1
1
2
3
4
3. How are the two graphs alike? (shape, growth/decay, focal point, etc.)
______________________________________________________________________________
______________________________________________________________________________
4. How are they different? (steepness, etc.)
______________________________________________________________________________
______________________________________________________________________________
 Complete the table of values for y  22 x and y  2 x
10 y
below, and draw both graphs on the right.
y  2x
y  22 x
x
x
9
8
y
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
y
7
6
5
4
3
2
1
x
-4
 This factor of change between the y-values of
x
( x / 2)
is known as a horizontal
y  2 and y  2
stretch. In this case, we have a horizontal stretch of 2.
 y  22 x has a horizontal stretch of ½ from y  2 x
4
-3
-2
-1
1
2
3
4
Date:________________________ Block:__________ Name:____________________________________
Part E- Vertical Translation
 Complete the table of values for ( y  3)  2 x and y  2 x below, and draw both graphs on the right.
y  2x
x
y
-3
-2
-1
0
1
2
3
10
( y  3)  2 x
x
y
-3
-2
-1
0
1
2
3
y
9
8
7
6
5
4
3
1. What is the focal point of ( y  3)  2 x ?
__________________________________
2. What is the horizontal asymptote of ( y  3)  2 x ?
2
1
x
-4
_______________________________________________
-3
-2
-1
1
2
3
4

Complete the table of values for ( y  2)  2 x and y  2 x
below, and draw both graphs on the right. (Remember to
make it calculator-friendly).
x
y2
( y  2)  2 x
x
y
x
y
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
3. What is the focal point of ( y  2)  2 x ?
__________________________________
4. What is the horizontal asymptote of ( y  2)  2 x ?
7
y
6
5
4
3
2
1
x
-4
-3
-2
-1
1
2
3
-1
-2
-3
_______________________________________________
 This change to the graph is called a vertical translation. The graph ( y  3)  2 x has a vertical
translation of 3. This means that the graph is vertically translated +3 units from the basic exponential
form. The graph ( y  2)  2 x has a vertical translation of -2. This means that the graph is vertically
translated -2 units from the basic exponential form.
5. What happens to the number in the brackets with the y term to get the vertical translation?
_____________________________________________________________________________________
5
4
Date:________________________
Part F- Horizontal Translations
Block:__________ Name:____________________________________

Complete the table of values for y  2( x  2) and y  2 x
below, and draw both graphs on the right.
x
y2
y  2( x  2)
x
y
x
y
-3
-5
-2
-4
-1
-3
0
-2
1
-1
2
0
3
1
10
y
9
8
7
6
5
4
3
2
( x  2)
1. What is the focal point of y  2
?
__________________________________
1
x
-4

( x 1)
Complete the table of values for y  2
and y  x
below, and draw both graphs on the right
x
y2
y  2( x 1)
x
y
x
y
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
4
-3
-2
-1
1
2
3
4
2
10
y
9
8
7
6
5
4
3
2. What is the focal point of y  2( x 1) ?
__________________________________
2
1
3. What do you notice about y  2( x 1) in comparison to y  2 x ?
-4
-3
-2
-1
1
2
3
_______________________________________
_____________________________________________________________________________________
4. Has the shape of the graph changed? __________________________________
5. What are the coordinates of the focal point of y  2( x 1) ? __________________________________
6. This change to the graph is called a horizontal translation. The graph y  2( x  2) has a horizontal
translation of –2 (i.e., the graph is translated horizontally –2 units from the basic exponential form).
6
x
4
Date:________________________
Part G- Mapping Rules
Block:__________ Name:____________________________________
 The Mapping Rule allows us to easily get a table of values for a transformed exponential function
without doing all of the mathematical computations
 The mapping rule that would represent this comparison is between the basic exponential y  b x and
1
( xh)
1
the transformed exponential ( y  k )  b c
would be ( x, y )  (()cx  h, ()ay  k ) where:
a
 Vertical stretch = a
 Horizontal stretch = c
 Vertical translation = +k
 Horizontal translation = +h
 There is a reflection in the x-axis if there is a negative in front of “a”
 There is a reflection in the y-axis if there is a negative in front of “c”
Example : Find the mapping Rule of the following exponential ( y  2)  3( x 3)
Vertical stretch = 1
Horizontal stretch = 1
Vertical translation = -2
Horizontal translation = -3
Reflection= No
Mapping Rule: x, y   x  3, y  2
Part H- Finding the focal point

To find the focal point of a transformed exponential, we simply use the mapping rule to transform the
focal point of y  b x (0,1)
Example : Find the focal point of the following exponential ( y  2)  3( x 3)
Mapping Rule: x, y   x  3, y  2
Focal point: 0,1  0  3, 1  2   3,1
7
Date:________________________ Block:__________ Name:____________________________________
Part I – Practice
For each exponential function:
1) State the transformations
2) The Mapping Rule.
3) Focal point
3) Create one table of values for the basic exponential and a second table of values for the given function (use
the mapping rule).
1
( x 1)
1
15
y
A)
( y  2)  3 2
2
10
5
x
-10
-5
5
10
-5
B) 2( y  1)  23( x  2)
5
y
x
-10
-5
5
-5
-10
-15
8
10