This document covers exponential functions including: - Representing exponential functions through tables, graphs, and equations. - Finding the domain, range, intercepts, zeros, and asymptotes of exponential functions. - Graphing exponential functions and describing how they increase or decrease based on the base value. - Transformations of exponential functions including reflections across axes and translations. - Properties of natural exponential functions where e is the base.

1. Exponential Functions
General Mathematics

2. Lesson Objectives
At the end of the lesson, the students must be
able to:
• represent an exponential function through its:
(a) table of values, (b) graph, and (c) equation;
• find the domain and range of an exponential
function;
• find the intercept, zeros, and asymptote of an
exponential function; and
• graph exponential functions.

3. Zero as an Exponent
• If a ≠ 0, then a⁰ = 1
Illustration:
To evaluate 3⁰ ∙ 3², we have
3⁰ ∙ 3² = 3⁰⁺² = 3² = 9 or
3⁰ ∙ 3² = 1 ∙ 3² = 1 ∙ 9 = 9

4. Negative Exponent
• If n is any integer, and a and b are not equal to
zero, then
• Illustration
Note: The negative exponent does not make the
answer negative
n
n
n
n
n
a
b
b
a
and
a
a
a 
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
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










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


 1
1
8
1
2
1
2 3
3

 


5. Definition
Transformation – the process of moving a figure
from the starting position to some ending
position without changing its size and shape.
Reflection – a transformation that produces a
new figure, which is a mirror image of the
original figure.

6. Definition
Translation – a shift or movement in a figure’s
location without changing its shape.
Glide Reflection – a combination of a reflection
and a translation (glide).

7. Example 1
Reflect each figure across the given axis.
a. b.
c. d.

8. Solution to Example 1
a. b.
c. d.

9. Example 2
Translate each shape in the direction indicated
by the arrow.
a. b.

10. Solution to Example 2
a. b.

11. Example 3
Translate each figure as indicated.
a. Translate the triangle 4 units to
the left and 3 units up.
b. Translate the parallelogram
3 units to the right and 2 units
down

12. Solution to Example 3
a. b.

13. Example 4
Perform a glide reflection on:
a. triangle MNP by translating the triangle
3 units to the right and 2 units up, and then
reflecting about the y-axis
b. the isosceles trapezoid LOVE by
translating the figure 2 units to the left
and 3 units up, followed by a reflection

14. Solution to Example 4
a. b.

15. Activity
The Exponent National High School with 1,500
population, including the teaching and non-
teaching staff, is located in one of the affected
by the earthquake. Due to recurrent
aftershcocks, everyone wants to know if
classes are suspended. The school principal
makes a decision and sends a text message to
the assistant principal and to the prefect of
activities. These two members of the
community each sends the text message to
two members of the community, and so on.

16. This texting diagram can be represented by a
tree diagram that goes like this:

17. 1. What do the smart phones of this tree
diagram represent? What do the segments
represent?
2. Based on the tree diagram, the number of
persons receiving the message is increasing.
a. Complete the next table to show the
number of persons receiving the message at a
given stage. Then, make a graph.

19. b. Describe how the number of persons receiving
the message increases as the texting stage
progresses. Use the graph from (a) to validate
your answer.
c. What is the required number of texting stages
needed to inform 1,500 persons?

20. Exponential Function
An exponential function can be written as
f (x) = bˣ
where b > 0, b ≠ 1, and x is any real number
In the equation f(x) = bˣ, b is a constant called
the base and x is an independent variable
called the exponent.

21. Here are some examples of exponential
functions.
The following are not exponential functions.

22. Properties of Exponential
Functions and Their Graphs
Let f(x) = bˣ, b > 1, and b ≠ 1.
1. The domain is the set of real numbers, (-∞,
∞).
2. The range is the set of positive real numbers,
(0, ∞).
3. If b > 1, f is an increasing exponential function.
If 0 < b < 1, f is a decreasing exponential
function

23. Example 5
Sketch the graph of y = 2ˣ, y = 3ˣ, and y = 4ˣ in
one plane. Describe the significance of the
constant b in the equation y = bˣ

24. Solution to Example 5
Compute some function values and list the results
in a table.

26. Example 6
Sketch the graph of y = 2ˣ, y = 2ˣ⁻¹ in one plane.
Describe the graph

27. Solution to Example 6
Compute some function values and list the results
in a table.

29. Example 7
Sketch the graph of y = 2ˣ, y = 2ˣ-1, and y = 2ˣ - 2
in one plane.
Describe the graphs.

30. Solution to Example 7
Compute some function values and list the results
in a table.

32. Example 8
Graph each group of functions in one plane.
Describe the graphs.
a. y = 2ˣ and y = -2ˣ
b. y = 2ˣ and y = 2⁻ˣ
c. y = 2ˣ, y = 2 (2ˣ), and y = 3(2ˣ)
d. y = 2ˣ and y = ½ (2ˣ)

33. Solution to Example 8

34. Solution to Example 8

35. Solution to Example 8

36. Solution to Example 8

37. Transformations Involving
Exponential Functions

38. Reflections in the Coordinate Axes
Reflections in the coordinate axes of the graph y
= f(x) are represented as follows:
1. Reflection in the x-axis: f(x) = -f(x)
2. Reflection in the y-axis: f(x) = f(-x)

39. Example 9
Use equation 1 to describe the transformation
that yields the graph of equation 2.

40. Solution to Example 9

41. Example 10
Graph: x = 2ʸ.

42. Solution to Example 10

43. The Euler’s number e is called the natural
number. The function f(x) = eˣ is called the
natural exponential junction. For the
exponential function f(x) = eˣ, e is the constant
2.71828183…, whereas x is the variable.

44. Example 11
Use a calculator to calculate the expression.
a. e⁰∙⁰¹
b. e.⁰∙⁵
c. e. ⁰∙¹
d. e²

45. Solution to Example 11

46. Example 12
Sketch the graph: y = eˣ and y = e⁻ˣ.

47. Solution to Example 12
a. Construct the table of values. Plot the points and
connect with smooth curve.

48. Solution to Example 12
b. Construct the table of values. Plot the points and
connect with smooth curve.

49. Example 13
Sketch the graph of each natural exponential
function.
a. f(x) = 2e⁻⁰∙²⁴ˣ
b. g(x) = e⁰∙⁵⁸ˣ
2
1

50. Solution to Example 13
To sketch the two graphs, use a calculator to construct a
table of values as shown below. After constructing the
table, plot the points and connect them with smooth
curve.

51. Example 14
Find the base of the exponential function
whose graph contains the given points.
a. (1, 4)
b. (2/3 , 4)

52. Solution to Example 14

53. Exercise A
Evaluate the following for the indicated value(s) of
x.
1. f(x) = 3x; x = 1, x =
2. f(x) = 4 - 3x; x = 4, x =
3. g(x) = 3ˣ; x = , x = 4
4. h(x) = ; x = , x = -1
3
1
2
1
x
2
3
1
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2
1
2
1

54. Exercise B
Make a table of coordinates then graph each
function.
1. f(x) = 5ˣ 6. g(x) = 4⁻ˣ⁺²
2. f(x) = 6⁻ˣ 7. h(x) =
3. g(x) = -5ˣ
4. f(x) = 3ˣ⁻² 8. f(x) =
5. f(x) =
1
2
1
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x
x
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3
2
x
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3
1

55. Exercise C
Find the base of the exponential function whose
graph contains the given points.
1. (2, 16) 5. (4, )
2. (1, 10)
3. (3, 64) 6. ( , 27)
4. (3, )
343
1
625
1
2
3