Exponential
Function
Prepared by: CARLEEN B. VILLAMIL
Complete the table below.
Activity:
Materials:
One 2- meter string and a pair of scissors.
a) At step 0, there is 1 string.
b) At step 1, fold the string intro two equal parts and
then cut at the middle. How many strings of equal
length do you have? Enter your answer in the table
below.
c) At step 2, again fold each of the strings equally and
then cut. How many strings of equal length do you have?
Enter your answer on the table below.
d) Continue the process until the table is completely
filled-up.
Step
No. Strings
0
1
1
2
3
4
5
6
7
Complete the table.
Step
0
No.
Strings
1
1
2
3
4
5
6
Complete the table.
Step
0
1
2
3
4
5
6
No.
Strings
1
2
4
8
16
32
64
• What pattern can be observed from
the data?
QUESTIONS
• Define a formula for the number of
strings as a function of the step
number
Definition:
Compount Interest
A starting amount of money (called the principal) can be invested at a
certain rate that is earned at the end of a given period of time (such as
one year). If the interest rate is compounded, the interest earned at
the end of the period is added to the princicpal, and this new amount
will earn interest in the next period. The same process is repeated for
each succeding period: interest previously earned will also earn interest
in the next period.
Example.
Mrs. de la Cruz invested P100,000, in a company that offers 6% interest
compounded annually. How much will this investment be worth at the end of
each year for the next five years?
Solution:
Let t be the time in years. Then we have:
Natural Exponential Function
While an exponential function may have various bases, a
frequently used based is the irrational number 𝒆 ≈ 𝟐. 𝟕𝟏𝟖𝟐𝟖 .
Because e is commonly used based, the natural exponential
function is defined having e as the base.
Exponential Functions, Exponential Equations,
and Exponential Inequalities
How are they similar? Or how are they different?
Definition
An exponential expression is an expression of the form
The definitions of exponential equations, inequalities and functions are shown below:
Exponential Equations Exponential
Inequality
Definition An Equation involving
exponential
expressions
Example
2𝑥−𝑥 2
7
1
=
343
An Inequation
involving
exponential
expressions
52𝑥 − 5𝑥+1 ≤ 0
Exponential
Functions
Function of the
form 𝑓 𝑥 =
𝑏 𝑥 , 𝑤ℎ𝑒𝑟𝑒 𝑏 >
0, 𝑏 ≠ 1
𝑓 𝑥 = (1.8)𝑥
or 𝑦 = (1.8)𝑥
Exponential Functions, Exponential Equations,
and Exponential Inequalities
• An exponential equation or inequality can be solved for all x values
that satisfy the equation or inequality. An exponential function
expresses a relationship between two variables (such as x and y) and
can be represented by a table of values or a graph.
Task 1
Determine whether the given is an exponential function, an exponential
equation, an exponential inequality, or none of these.
Questions:
• How did you know that the given equation is exponential function?
• How did you know that the given equation is exponential equation?
• How did you know that the given equation is exponential inequalities?
• How does the exponential equation differ from exponential
inequalities?
Solving exponential equations and Inequalities
RECALL:
Which of the following are exponential equations? Exponential
inequalities? Neither?
Solving exponential equations
One-to-one Property of Exponential Functions.
Our strategy to solve exponential equations is to write both sides of the
equation as powers of the same base.
Solution:
Seatwork!
Solve for x.
Evaluation:
Solve for x.
Solving exponential inequalities
Application
SEATWORK 2:
Solve for x.
1.
2.
GRAPHING
EXPONENTIAL
FUNCTIONS
INTRODUCTION
The graph of exponential function is a
necessary tool in describing its bahavior
and characteristics –its intercepts,
asymptotes and zeros. A graph can also
provide insights as to real-life situations
that can be modeled by exponential
functions.
Example
It can be1.observed that the𝑥 function is defined for all values of x,
Sketch the graph of 𝑓 𝑥 = 2
is strictly increasing, and attains only positive y-values. As x
decreases
without
bound,
the function
approaches 0. That is, the
Step
1. Construct
a table
of values
of
ordered
the given
function. The
line is apairs
horizontal
asymptote.
table of value for is as follows:
Step 2. Plot the points found in the table
and connect them using a smooth curve.
It can be observed that the function is defined for all values of x, is strictly
Example 2.
decreasing, and attains only positive values. As x increases without bound,
the function approaches 0. That
is, the line is a horizontal asymptote.
1 𝑥
Sketch the graph of g 𝑥 = ( )
2
Step 1. The table of values for g(x) is a
follows:
Step 2. Plot the points found in the table
and connect them using a smooth curve.
EVALUATION