Exponential Functions L. Waihman 2002
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 increasing, one-to-one functions. • 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 function is neither even nor odd. There is no symmetry. • There is no local extrema.
More Characteristics of • The domain is • The range is • End Behavior: • As • As • The y-intercept is • The horizontal asymptote is • There is no x-intercept. • There are no vertical asymptotes. • This is a continuous function. • It is concave up.
Domain: Range: Y-intercept: • How would you graph Horizontal Asymptote: Inc/dec? increasing Concavity? up • How would you graph Domain: Range: Y-intercept: Horizontal Asymptote: Inc/dec? increasing up Concavity?
Recall that if then the graph of is a reflection of about the y-axis. • Thus, if then Domain: Range: Y-intercept: Horizontal Asymptote: Concavity? up
How would you graph • Notice that the reflection is decreasing, so the end behavior is: Is this graph increasing or decreasing? Decreasing.
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. Reflect @ x-axis Vertical stretch 3 Vertical shift down 1 Vertical shift up 3
More Transformations Reflect about the x-axis. Vertical shrink ½ . Horizontal shift left 2. Horizontal shift right 1. Vertical shift up 1. Vertical shift down 3. Domain: Domain: Range: Range: Horizontal Asymptote: Horizontal Asymptote: Y-intercept: Y-intercept: Inc/dec? decreasing Inc/dec? increasing Concavity? down Concavity? up
The number e • The letter e is the initial of the last name of Leonhard Euler (1701-1783) • 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 2.7182818284590452. • The function is called the Natural Exponential Function
Domain: Range: Y-intercept: H.A.: Continuous Increasing No vertical asymptotes and
Transformations Vertical stretch 3. Horizontal shift left 2. Reflect @ x-axis. Vertical shift up 2 Vertical shift up 2. Vertical shift down 1. Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Inc/dec? increasing Inc/dec? increasing Inc/dec? decreasing Concavity? up Concavity? up Concavity? down
Exponential Equations • Equations that contain one or more exponential expressions are called exponential equations. • Steps to solving some exponential equations: • Express both sides in terms of same base. • When bases are the same, exponents are equal. i.e.:
Exponential Equations • Sometimes it may be helpful to factor the equation to solve: or There is no value of x for which is equal to 0.
Exponential Equations • Try: • 1) 2) or