Exponential Functions L. Waihman 2002

A function that can be expressed in the form A function that can be expressed in the form and is positive, is called an Exponential Function. and is positive, is called an Exponential Function. Exponential Functions with positive values of x are increasing, one-to-one functions. Exponential Functions with positive values of x are increasing, one-to-one functions. The parent form of the graph has a y-intercept at (0,1) and passes through (1,b). The parent form of the graph has a y-intercept at (0,1) and passes through (1,b). The value of b determines the steepness of the curve. The value of b determines the steepness of the curve. The function is neither even nor odd. There is no symmetry. The function is neither even nor odd. There is no symmetry. There is no local extrema. There is no local extrema.

The domain is The domain is The range is The range is End Behavior: End Behavior: As As The y-intercept is The y-intercept is The horizontal asymptote is The horizontal asymptote is More Characteristics of There is no x-intercept. There is no x-intercept. There are no vertical asymptotes. There are no vertical asymptotes. This is a continuous function. This is a continuous function. It is concave up. It is concave up.

How would you graph How would you graph Domain: Range: Y-intercept: Domain: Range: Y-intercept: Inc/dec? Horizontal Asymptote: Horizontal Asymptote: Concavity?  How would you graph up increasing up

Recall that if then the graph of is a reflection of about the y-axis. Recall that if then the graph of is a reflection of about the y-axis. Thus, if then Thus, if then Domain: Range: Y-intercept: Horizontal Asymptote: Concavity? up

Notice that the reflection is decreasing, so the end behavior is: Notice that the reflection is decreasing, so the end behavior is: Is this graph increasing or decreasing? Decreasing.  How would you graph

How does b affect the function? If b>1, then f is an increasing function, and If 0<b<1, then f is a decreasing function, and

Transformations Exponential graphs, like other functions we have studied, can be dilated, reflected and translated. It is important to maintain the same base as you analyze the transformations. Vertical shift up 3 x-axis Vertical stretch 3 Vertical shift down 1

More Transformations Reflect about the x-axis. Horizontal shift right 1. Vertical shift up 1. Vertical shrink ½. Horizontal shift left 2. Vertical shift down 3. Domain: Range: Y-intercept: Horizontal Asymptote: Inc/dec? Concavity? Domain: Range: Y-intercept: Horizontal Asymptote: Inc/dec? Concavity? decreasing down increasing up

The number e The letter e is the initial of the last name of Leonhard Euler ( ) who introduced the notation. Since has special calculus properties that simplify many calculations, it is the natural base of exponential functions. The value of e is defined as the number that the expression approaches as n approaches infinity. The value of e to 16 decimal places is The function is called the Natural Exponential Function

Domain: Range: Y-intercept: H.A.: Continuous Increasing No vertical asymptotes and

Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Transformations Vertical stretch 3. Vertical shift up 2. x-axis. Vertical shift down 1. Horizontal shift left 2. Vertical shift up 2 Inc/dec?increasing Concavity?up Inc/dec? decreasing Concavity?down increasing Concavity?up

Exponential Equations Equations that contain one or more exponential expressions are called exponential equations. Equations that contain one or more exponential expressions are called exponential equations. Steps to solving some exponential equations: Steps to solving some exponential equations: 1. Express both sides in terms of same base. 2. When bases are the same, exponents are equal. i.e.: i.e.:

Exponential Equations Sometimes it may be helpful to factor the equation to solve: Sometimes it may be helpful to factor the equation to solve: There is no value of x for which is equal to 0. or

Exponential Equations Try: Try: 1) 2) 1) 2) or