**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**

a^{s}*a^{t}=a^{s+t}

(a^{s})^{t}=a^{st}

(ab)s=as*bs

1^{s}=1

a^{-s}=1/a^{s}=(1/a)^{s}

a^{0}=1

**Objective 2:** Graphing Exponential Functions

**Build a Table of Values**:

f(x)=3^{x}

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)=3
^{x}get closer and closer to zero

**Properties of the Exponential Function f(x)=a ^{x}, 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)=-a
^{x}→ reflect over x-axis - f(x)=a
^{-x}→ reflect over y-axis - f(x)=a
^{x}-b → vertical shift “b” units- If “b” is negative (-), shift “b” units down
- If “b” is positive (+), shift “b” units up

- f(x)=a
^{x-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)=3^{x} 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)=e^{x}

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:**

4^{x-3}=16

1.Since 16=4^{2}, the equation can be rewritten as 4^{x-3}=16=4^{2}

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. 3^{5.3}

2. 7^{1.2}

3. 9^{1.425}

4. 6^{3.14}

5. e^{2.7}

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

6. f(x)=6^{x}+32

7. f(x)=6^{x+6}

8. f(x)=7+8(3^{x})

9. f(x)=3-7e^{4x}

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)=a^{x}:

11. f(x)=4^{x}+32

12. f(x)=(½)^{1-x}

13. f(x)=3^{3/x}

14. f(x)=-4^{x-3}-7

15. f(x)=7^{4-3x}+3

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

16. 7^{x-4}=16,807

17. 13^{x-1}=28,561

18. 3^{x*2}=9

19. 5^{x/2}=25

20. 8^{2x}=1,073,741,824