6.3 Exponential Functions

Slides:

Advertisements

Similar presentations

Growth and Decay Exponential Models. Differs from polynomial functions. Polynomial Functions have exponents of whole numbers Exponential Functions have.

Advertisements

6.3 Exponential Functions

State the domain and range of each function. 3.1 Graphs of Exponential Functions.

Exponential Functions Brought to you by Tutorial Services The Math Center.

Unit 3 Functions (Linear and Exponentials)

Unit 3 Functions (Linear and Exponentials)

Logarithmic Functions. Definition of a Logarithmic Function For x > 0 and b > 0, b = 1, y = log b x is equivalent to b y = x. The function f (x) = log.

Logarithmic Functions Section 2. Objectives Change Exponential Expressions to Logarithmic Expressions and Logarithmic Expressions to Exponential Expressions.

Intercepts, Exponentials, and Asymptotes Section 3.4 Standard: MCC9-12.F.IF.7a&e Essential Question: How do you graph and analyze exponential functions.

1 6.3 Exponential Functions In this section, we will study the following topics: Evaluating exponential functions with base a Graphing exponential functions.

Exponential Functions Section 1. Exponential Function f(x) = a x, a > 0, a ≠ 1 The base is a constant and the exponent is a variable, unlike a power function.

Transformations xf(x) Domain: Range:. Transformations Vertical Shifts (or Slides) moves the graph of f(x) up k units. (add k to all of the y-values) moves.

How does one Graph an Exponential Equation?

4.2 Logarithmic Functions

Exponential and Logarithmic Functions and Equations

Section 6.3. This value is so important in mathematics that it has been given its own symbol, e, sometimes called Euler’s number. The number e has many.

9/18/ : Parent Functions1 Parent Functions Unit 1.

Aim: What is an exponential function?

Exponential Functions

Exponential Functions Section 1. Exponential Function f(x) = a x, a > 0, a ≠ 1 The base is a constant and the exponent is a variable, unlike a power function.

Exponential Functions L. Waihman A function that can be expressed in the form A function that can be expressed in the form and is positive, is called.

Section 6.3 – Exponential Functions Laws of Exponents If s, t, a, and b are real numbers where a > 0 and b > 0, then: Definition: “a” is a positive real.

Section 4.1 Exponential Functions

Sullivan Algebra and Trigonometry: Section 5.3 Exponential Functions Objectives of this Section Evaluate Exponential Functions Graph Exponential Functions.

1 Factoring Practice (5 questions). 2 Factoring Practice (Answers)

Day 6 Pre Calculus. Objectives Review Parent Functions and their key characteristics Identify shifts of parent functions and graph Write the equation.

6.3 Logarithmic Functions. Change exponential expression into an equivalent logarithmic expression. Change logarithmic expression into an equivalent.

State the domain and range of each function Exponential Growth and Decay.

Exponential Functions What You Will Learn How to graph exponential functions And how to solve exponential equations and inequalities.

Exponential Functions and Their Graphs 2 The exponential function f with base a is defined by f(x) = a x where a > 0, a  1, and x is any real number.

8-2: Exponential Decay Objective Ca Standard 12: Students know the laws of fractional exponents, understand exponential functions and use these functions.

Real Exponents Chapter 11 Section 1. 2 of 19 Pre-Calculus Chapter 11 Sections 1 & 2 Scientific Notation A number is in scientific notation when it is.

Exponential Functions Evaluate Exponential Functions Graph Exponential Functions Define the number e Solve Exponential Equations.

9.1 Exponential Functions

5.4 Logarithmic Functions. Quiz What’s the domain of f(x) = log x?

SECTION 4.3 EXPONENTIAL FUNCTIONS EXPONENTIAL FUNCTIONS.

Exponential Functions

Exponential & Logarithmic functions. Exponential Functions y= a x ; 1 ≠ a > 0,that’s a is a positive fraction or a number greater than 1 Case(1): a >

EXPONENTIAL FUNCTIONS Section TOPIC FOCUS I can… Identify exponential growth and decay Graph exponential functions.

Exponential Functions Exponential Growth Exponential Decay y x.

Graphing Exponential function parent function: y = 2 x X is the exponent!!! What does this look like on a graph? In the parent function the horizontal.

 A function that can be expressed in the form and is positive, is called an Exponential Function.  Exponential Functions with positive values of x are.

Exponential Growth Exponential Decay Example 1 Graph the exponential function given by Solution xy or f(x) 0 1 –1 2 – /3 9 1/9 27.

(a) (b) (c) (d) Warm Up: Show YOUR work!. Warm Up.

Section 1.4 Transformations and Operations on Functions.

Exponential & Logarithmic functions. Exponential Functions y= a x ; 1 ≠ a > 0,that’s a is a positive fraction or a number greater than 1 Case(1): a >

Math – Exponential Functions

Section Vocabulary: Exponential function- In general, an equation of the form, where, b>0, and, is known as an exponential function. Exponential.

3.1 Exponential Functions and Their Graphs Objectives: Students will recognize and evaluate exponential functions with base a. Students will graph exponential.

LEQ: How do you evaluate logarithms with a base b? Logarithms to Bases Other Than 10 Sec. 9-7.

Graphs of Exponential Functions. Exponential Function Where base (b), b > 0, b  1, and x is any real number.

Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and.

1. Given the equation y = 650(1.075)x

Section 6.2 – Graphs of Exponential Functions

Sullivan Algebra and Trigonometry: Section 6.3 Exponential Functions

9.6 Graphing Exponential Functions

How does one Graph an Exponential Equation?

Exponential Functions

MATH 1310 Section 5.1.

Graphing Exponential Functions Exponential Growth p 635

Graphing Exponential Functions

6.9 Graphing Exponential Equations

MATH 1310 Section 5.1.

Sullivan Algebra and Trigonometry: Section 6.2

7.4 Graphing Exponential Equations

Exponential Functions and Their Graphs

MATH 1310 Section 5.1.

Exponential Functions and Their Graphs

6.3 Exponential Functions

Presentation transcript:

6.3 Exponential Functions MAT 204 SPRING 2009 6.3 Exponential Functions In this section, we will study the following topics: Evaluating exponential functions with base b Graphing exponential functions with base b

So, in an exponential function, the variable is in the exponent.

Exponential Functions Which of the following are exponential functions?

Graphs of Exponential Functions Just as the graphs of all quadratic functions have the same basic shape, the graphs of exponential functions have the same basic characteristics. They can be broken into two categories— exponential growth exponential decay (decline)

The Graph of an Exponential Growth Function We will look at the graph of an exponential function that increases as x increases, known as the exponential growth function. It has the form Example: f(x) = 2x x f(x) -5 -4 -3 -2 -1 1 2 3 f(x)=2x Notice the rapid increase in the graph as x increases The graph increases slowly for x < 0. y-intercept is (0, 1) Horizontal asymptote is y = 0.

The Graph of an Exponential Decay (Decline) Function We will look at the graph of an exponential function that decreases as x increases, known as the exponential decay function. It has the form Example: g(x) = 2-x g(x)=2-x x f(x) -3 -2 -1 1 2 3 4 5 Notice the rapid decline in the graph for x < 0. The graph decreases more slowly as x increases. y-intercept is (0, 1) Horizontal asymptote is y = 0.

Graphs of Exponential Functions Notice that f(x) = 2x and g(x) = 2-x are reflections of one another about the y-axis. Both graphs have: y-intercept (___,___) and horizontal asymptote y = . The domain of f(x) and g(x) is _________; the range is _______.

Graphs of Exponential Functions MAT 204 SPRING 2009 Graphs of Exponential Functions Also, note that , using the properties of exponents. So an exponential function is a DECAY function if The base a is greater than one and the function is written as f(x) = b-x -OR- The base a is between 0 and 1 and the function is written as f(x) = bx

Graphs of Exponential Functions MAT 204 SPRING 2009 Graphs of Exponential Functions Examples: In this case, a = 0.25 (0 < a < 1). In this case, a = 5.6 (a > 1).

Transformations of Graphs of Exponential Functions Look at the following shifts and reflections of the graph of f(x) = 2x. The new horizontal asymptote is y = 3

Transformations of Graphs of Exponential Functions

Transformations of Graphs of Exponential Functions Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions. State the domain, range and the horizontal asymptote for each. 1.

Transformations of Graphs of Exponential Functions Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions. State the domain, range and the horizontal asymptote for each. 2.

Transformations of Graphs of Exponential Functions Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions. State the domain, range and the horizontal asymptote for each. 3.

Transformations of Graphs of Exponential Functions Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions. State the domain, range and the horizontal asymptote for each. 4.

Graph using transformations and determine the domain, range and horizontal asymptote. B) A) D) C)