OBJECTIVES
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
as*at=as+t
(as)t=ast
(ab)s=as*bs
1s=1
a-s=1/as=(1/a)s
a0=1
Objective 2: Graphing Exponential Functions
Build a Table of Values:
f(x)=3x
X | Y |
---|---|
-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
Example:
f(x)=3(-x+6)-2
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
Graph:
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
e=2.718281828459…
f(x)=ex
Graph:
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
Example:
Solve f(x)=2(x-4)+7 for the the value y=8
Graph:
Intersection: (4,8)
Solution: when y=8, x=4
Analytically:
If au=av, then u=v
Example:
4x-3=16
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