4.2 Exponential Functions and Equations Simplify expressions and equations involving rational exponents. Explore Exponential functions and their graphs.

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 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.

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

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

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

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 Vertical shrink Horizontal shift Horizontal shift Vertical shift Vertical shift Domain: Domain: Range: Range: Horizontal Asymptote: Horizontal Asymptote: Y-intercept: Y-intercept: Inc/dec? Inc/dec? Concavity? Concavity?

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 Reflect @ x-axis. Horizontal shift left 2. Vertical shift Vertical shift down 1. Vertical shift up 2 Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Inc/dec? Inc/dec? Inc/dec? Concavity? Concavity? Concavity?

Exponential Equations: The Rules of Exponents

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:

Exponential Equations Try: 1) 2)

Interest Problems 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 If you deposit $10,000 in an account that pays 4% interest compounded annually, how much money will you have in your account at the end of 15 years? Write an exponential function that represents this situation. 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡

Interest Problems – part 2 If you deposit $10,000 in an account that pays 4% interest compounded quarterly, how much money will you have in your account at the end of 15 years? Write an exponential function that represents this situation. 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡

Interest Problems – part 3 If you deposit $10,000 in an account that pays 4% interest compounded continuously, how much money will you have in your account at the end of 15 years? Write an exponential function that represents this situation. 𝐴=𝑃 𝑒 𝑟𝑡

Exponential Growth and Decay Problems In 1995, there were 85 rabbits in Central Park.  The population increased by 12% each year.  How many rabbits were in Central Park in 2005?

In 2000, 50 grams of radium were stored In 2000, 50 grams of radium were stored.  The half-life of radium is 1,620 years.  How many grams of radium remains after 4860 years?  Remember, half-life is the amount of time it takes for half of the amount of a substance to decay.