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. The value of b determines the steepness of the curve and whether there is growth or decay. The value of b determines the steepness of the curve and whether there is growth or decay. If a > 1, there is a vertical stretch, If a > 1, there is a vertical stretch, if 0<a<1, there is a vertical compression (shrink), if 0<a<1, there is a vertical compression (shrink), if a is negative, the graph reflects over the x axis. if a is negative, the graph reflects over the x axis. h moves the graph left and right, h moves the graph left and right, k moves it up and down k moves it up and down F(x) = ab x-h +k

If a is positive If a is positive The domain is The domain is The range is (k, +inf) The range is (k, +inf) The horizontal asymptote The horizontal asymptote is y = k is y = k More Characteristics of F(x) = ab x-h +k (

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

If a is positive and… If a is positive and… If b > 1, the function is increasing If b > 1, the function is increasing This is called exponential growth This is called exponential growth If 0< b < 1, the function is decreasing If 0< b < 1, the function is decreasing This is called exponential decay This is called exponential decay Domain: Range: Y-intercept: Horizontal Asymptote: x=0 F(x) = ab x-h +k

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

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 Reflect over the 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

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