Chapter 3: Exponential, Logistic, and Logarithmic Functions 3 Chapter 3: Exponential, Logistic, and Logarithmic Functions 3.1a &b Homework: p. 286-287 1-39 odd

Overview of Chapter 3 So far in this course, we have mostly studied algebraic functions, such as polys, rationals, and power functions w/ rat’l exponents… The three types of functions in this chapter (exponential, logistic, and logarithmic) are called transcendental functions, because they “go beyond” the basic algebra operations involved in the aforementioned functions…

Consider these problems: Evaluate the expression without using a calculator. 1. 2. 3. 4.

We begin with an introduction to exponential functions: First, consider: Now, what happens when we switch the base and the exponent ??? This is a familiar monomial, and a power function… one of the “twelve basics?” This is an example of an exponential function

Definition: Exponential Functions   Let a and b be real number constants. An exponential function in x is a function that can be written in the form where a is nonzero, b is positive, and b = 1. The constant a is the initial value of f (the value at x = 0), and b is the base. Note: Exponential functions are defined and continuous for all real numbers!!!

Identifying Exponential Functions Which of the following are exponential functions? For those that are exponential functions, state the initial value and the base. For those that are not, explain why not. 1.   4. Initial Value = 1, Base = 3 Initial Value = 7, Base = 1/2 2. Nope!  g is a power func.!   5. 3.   Nope!  q is a const. func.! Initial Value = –2, Base = 1.5

More Practice with Exponents Given , find an exact value for: 1. 2. 3. 4. 5.

Finding an Exponential Function from its Table of Values Determine the formula for the exp. func. g:   x g(x) –2 4/9 x 3 Initial Value: –1 4/3 x 3 4 x 3 1 12   x 3 2 36 The Pattern? Final Answer:  

Finding an Exponential Function from its Table of Values Determine the formula for the exp. func. h: General Form:   x h(x) Initial Value: –2 128 x 1/4 Solve for b: –1 32 x 1/4   8 x 1/4 1 2 Final Answer: x 1/4 2 1/2 The Pattern?

How an Exponential Function Changes (a recursive formula)   If a > 0 and b > 1, the function f is increasing and is an exponential growth function. The base b is its growth factor. If a > 0 and b < 1, f is decreasing and is an exponential decay function. The base b is its decay factor. Does this formula make sense with our previous examples?

Graphs of Exponential Functions

We start with an “Exploration” Graph the four given functions in the same viewing window: [–2, 2] by [–1, 6]. What point is common to all four graphs? Graph the four given functions in the same viewing window: [–2, 2] by [–1, 6]. What point is common to all four graphs?

We start with an “Exploration” Now, can we analyze these graphs???

Exponential Functions f(x) = b Domain: Continuity: Continuous Range: Symmetry: None Boundedness: Below by y = 0 Extrema: None H.A.: y = 0 V.A.: None If b > 1, then also If 0 < b < 1, then also f is an increasing func., f is a decreasing func.,

Definition: The Natural Base e In Sec. 1.3, we first saw the “The Exponential Function”: Natural (we now know that it is an exponential growth function  why?) But what exactly is this number “e”??? Definition: The Natural Base e

Analysis of the Natural Exponential Function The graph: Domain: All reals Range: Continuous Increasing for all x No symmetry Bounded below by y = 0 No local extrema H.A.: y = 0 V.A.: None End behavior:

Guided Practice Describe how to transform the graph of f into the graph of g. 1. Trans. right 1 2. Reflect across y-axis 3. Horizon. shrink by 1/2 4. Reflect across both axes, Trans. right 2 5. Reflect across y-axis, Vert. stretch by 5, Trans. up 2

Guided Practice Determine a formula for the exponential function whose graph is shown.

Whiteboard… Exponential Decay Exponential Growth State whether the given function is exp. growth or exp. decay, and describe its end behavior using limits. Exponential Decay Exponential Growth

Whiteboard… x > 0 x > 0 Solve the given inequality graphically. The graph? x > 0 The graph? x > 0