Exponential Function

Objectives:
1. Define an exponential
function; and
2. Identify an exponential
function through its (a)
table of values, (b) graph
and (c) equation.

Moral of the Story:
“Understanding the value of
small things as a foundation
of doing great things.”

Group Activity

Situation 1:
Shown at the left is a
pattern of “growing”
squares from match
sticks.
1.Study the pattern and draw a picture of the next
likely figure in the pattern.
2.Make a table of values showing the number of
matchsticks on one side of the figure and the
perimeter of the figure.
3.Sketch the graph of the relationship shown in
the table (side length, perimeter)
4.Describe in words the relationship between the
perimeter and the side of each figure formed.
No. of matchsticks on one
side of the figure
1 2 3 4 5 6
Perimeter of the figure 4 8 12

Situation 2:
Shown at the left is a
pattern of “growing”
squares from match
sticks.
1. Study the pattern and draw a picture of the
next likely shape in the pattern.
2. Make a table of values showing the number of
matchsticks on one side of the figure and the
area of the figure in square units.
3. Sketch the graph of the relationship shown in
the table (side length, area)
4. Describe in words the relationship between
the area and the side of each figure formed.
No. of matchsticks on one
side of the figure
1 2 3 4 5 6
Area of the figure
in square units
1 4 9

Situation 3: Shown at
the left is a pattern
of “growing” squares
from match sticks
where the length of
the side was doubled
every time.
1. Study the pattern and draw a picture of the
next likely shape in the pattern.
2. Make a table of values showing the area of
each square figure in square units.
3. Sketch the graph of the relationship shown in
the table (order of square figure, area)
4. Describe in words the relationship between
the area and the order of square figures.
Order of square figures 1 2 3 4 5 6
Area of the figure
in square units
1 4 16

Discussion
Solution for #1
Solution for #2
Solution for #3

On the story of a chessboard, the
pattern of increasing the number of
wheat grains on each square of a chess
board is presented by the table below:
Noticed that the number of grains
for each squares can be broken down as
a power of 2.
Number of squares 1 2 3 4 5 6 7 8…
Number of grains 1 2 4 8 16 32 64
128
…

Order of
squares
No. of grains Breakdown of y as a power of 2
1 1 = 20
2 2 2 =21
3 4 2  2 =22
4 8 2  2  2 =23
5 16 2  2  2  2 =24
6 32 2  2  2  2  2 =25
7 64 2  2  2  2  2  2 =26
8 128 2  2  2  2  2  2  2 =27
: :
64 9.22337 x 1018 2  2  2  2  2  2  …  2 =263
: :
x ∞ 2  2  2  2  …  2 = 2x

The graph of
f(x) = 2x
x - values -3 -2 -1 0 1 2 3 4
f(x) 1/8 1/4 1/2 1 2 4 8 16

Definition:
• An exponential function can be
written as
• Where b > 0, b  1, and x is any real
numbers
power
base
exponent

The graph of
f(x) = 2x
x-values -3 -2 -1 0 1 2 3 4
f(x) 1/8 1/4 1/2 1 2 4 8 16
The graph of
f(x) = (1/2)x
x-values -3 -2 -1 0 1 2 3 4
f(x) 8 4 2 1 1/2 1/4 1/8 1/16

Important Notes:
We identify exponential function by
a. Table of Values – if there is no equal
differences in terms of y.
b. Graph – if b > 1, the graph is increasing
rapidly to the right and coming
closer and closer to the x-axis to
the left.
- if 0 < b < 1, the graph is increasing
rapidly to the left and coming
closer and closer to the x-axis to
the right.
c. Equation – the variable is on the exponent.

Check This Out:
Consider a game
called Tower of Hanoi.
It consists of circular discs of graduated sizes
inserted in one of the three pegs. The problem is to
transfer the discs to any of the other pegs in the
fewest number of possible moves with no bigger
disk placed above a smaller disk and a disks can
only be moved one at a time.
a. Complete the table below.
b. Sketch the graph of the relationship shown on
the table (x,y).
c. Find the relation between x and y.Number of Disks (x) 1 2 3 4 5 6 7
No. of Moves (y) 1 3 63 127

Generalization

Application:
1. Determine whether the following
table of values describes an
exponential function or not.
a.
b.
c.
d.
x -2 -1 0 1 2 3
f(x) -4 -2 0 2 4 6
x 0 1 2 3 4 5
f(x) 1 3 9 27 81 243
x -2 -1 0 1 2 3
f(x) 12 3 0 3 12 27
x 0 1 2 3 4 5
f(x) 1 4 16 64 256 1024
Not exponential
Not exponential
Exponential
Exponential

Application:
graph is an exponential function or
not.
a. b.
c. d.
Not exponential
Not exponential
Exponential
Exponential

Application:
function is an exponential function
or not.
a. f(x) = -3x b. f(x) = (-3)x - 3
c. f(x) = x2 + 1 d. f(x) = 2/x
e. f(x) = 4x+3 f. f(x) = 4xx
g. f(x) = (x-2)-1 h. f(x) = 8(-x)
Exponential Not exponential
Not exponentialExponential
Not exponential Not exponential
Not exponential Exponential

Evaluation:
1. Tell whether each statement is
True or False.
a. f(x) = 3x2 is an exponential function.
b. If f(x) = 2x, then f(-1) = 2.
c. If f(x) = (1/2)x, then f(-1) = 2.
d. The function f(x) = 1x is an
exponential function.
e. The function f(x) = (-3)x is an
exponential function.

Evaluation:
2. Determine whether the given
function represents an exponential
function or not.
a. y – 7 = x3 + 3 b. 5y = 43x
c. 2x–4 = y + 3 d. 3 + 5x+2 = 4y

Evaluation:
3. Given the following tables of
ordered pairs. Identify those which
exhibit an exponential trend.
a. b.
c. d.
e. f.
x 1 2 3 4
y 5 25 125 625
x -3 -2 -1 0
y 6 4 2 0
x -1 0 1 2
y -1 0 1 8
x 1 2 3 4
y 6 18 54 162
x 3 4 5 6
y 1/8 1/16 1/32 1/64
x 4 3 2 1
y 16/3 8/3 4/3 2/3

Exponential Function

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