1. Evaluate Exponential Functions
2. Graphing Exponential Functions
3. Transformations on Exponential Functions
4. Defining the Number “e”
5. Solving Exponential Functions

Objective 1: Evaluating Exponential Functions

Standard form of an Exponential Function: f(x)=ax

  • “a” is a positive real number not equal to 1
    • “a” is also called the base
  • Domain: All real numbers
  • Range: All positive real numbers

Properties of Exponents







Objective 2: Graphing Exponential Functions

Build a Table of Values:

-3 3(-3)=1/27
-2 3(-2)=1/9
-1 3(-1)=1/3
0 3(0)=1
1 3(1)=3
2 3(2)=9
3 3(3)=27

Plot the Points:


Important Details about the Graph:

  • No x-intercepts
  • y-intercept is (0,1)
  • The data states that as x approaches ∞, the values of f(x)=3x get closer and closer to zero

Properties of the Exponential Function f(x)=ax, a>1

1. Domain: (-∞,∞)

2.Range: (0,∞)

3. There are no x-intercepts

4. The y-intercept is (1,0)

5. The x-axis (y=0) is a horizontal asymptote as x→∞

6. The graph of f contains the points (0,1), (1,a), and (-1, 1/a)

7. The general look of an exponential function is smooth and continuous with no corners of gaps


Objective 3: Transformations of Exponential Functions
When the exponential function isn’t in standard form, transformations are used to graph the function:

  • f(x)=-ax → reflect over x-axis
  • f(x)=a-x → reflect over y-axis
  • f(x)=ax-b → vertical shift “b” units
    • If “b” is negative (-), shift “b” units down
    • If “b” is positive (+), shift “b” units up
  • f(x)=ax-b → horizontal shift “b” units
    • If “b” is negative (-), shift “b” units right
    • If “b” is positive (+), shift “b” units left


List the Transformations:
1.Graph f(x)=3x using a table of values
2.Reflect over the y-axis
3.Vertical shift down 2 units
4.Horizontal shift left 6 units



Objective 4: Defining the number “e

Many problems that occur in nature that can be modelled by an exponential function and require the use of a base that is a certain irrational number, “e

1.Names: e constant or Euler's number

2.Mathematical constant that is both a real and irrational number

3.e is similar to π




Objective 5: Solving Exponential Functions

Graphically: Intersection of graphs

1.Graph the given exponential equation

2.Graph the linear x or y value that you are solving for

3.If solving for x, the x value in the (x,y) coordinate of the intersection point is the solution

4.If solving for y, the y value in the (x,y) coordinate of the intersection point is the solution


Solve f(x)=2(x-4)+7 for the the value y=8


Intersection: (4,8)
Solution: when y=8, x=4

If au=av, then u=v



1.Since 16=42, the equation can be rewritten as 4x-3=16=42

2.With the same base on each side (4), the above property can be applied→ x-3=2, x=5

Practice Problems:

For problems 1-5 approximate each number with a calculator:

1. 35.3
2. 71.2
3. 91.425
4. 63.14
5. e2.7

For Problems 6-10 use the given equations to graph each function. Also state the domain and range of each function:

6. f(x)=6x+32
7. f(x)=6x+6
8. f(x)=7+8(3x)
9. f(x)=3-7e4x
10. f(x)=e-x

For Problems 11-15 use the given equations, graph a model of the function, and list the transformations on the base function f(x)=ax:

11. f(x)=4x+32
12. f(x)=(½)1-x
13. f(x)=33/x
14. f(x)=-4x-3-7
15. f(x)=74-3x+3

For Problems 16-20, solve each exponential equation analytically:

16. 7x-4=16,807
17. 13x-1=28,561
18. 3x*2=9
19. 5x/2=25
20. 82x=1,073,741,824