a = c b Exponent Base Solution Rational Exponents Irrational Exponents We have worked with ______________ _______________ in the past, but in this section will begin to look at __________________ ____________________. In order to put irrational exponents into the calculator, use the ______________ key. Rational Exponents Irrational Exponents Caret

0.117 0.113 5.900 0.465 3640.950 Examples: 2-3.1 = _____________ 2-∏ = _____________ 125/7 = ___________ (0.6) 3/2 = _____________ e 8.2 = _____________ 0.117 0.113 5.900 0.465 3640.950

What is e? Much like π, e is a special irrational number used in math It occurs in natural exponential growth and decay situations It was discovered by a mathematician Leonard Euler Called the natural base e or Euler’s number

Exponential functions 𝒚= 𝒂 𝒙 Exponential expressions can also be written as ___________________ ________________, which take the form _______________________. Let’s look at the graphs formed by a standard exponential function: Exponential functions 𝒚= 𝒂 𝒙 𝟎<𝒂<𝟏 𝒂>𝟏

reflection translation Form of the Function Changes to the Graph When the graph of the exponential function 𝑦= 𝑎 𝑥 is modified, the rules of ______________ and ___________________ should be applied. reflection translation Form of the Function Changes to the Graph y = a (x-c)   y = a (x+c) y = a x - c y = a x + c y = - a x y = a - x move right c units move left c units move down c units move up c units Reflected in x axis Reflected in y axis

Exponential functions can also be applied to ideas of compound interest using the formulas __________________________ for compounding periodically and __________________________ for compounding continuously. 𝑨=𝑷 (𝟏+ 𝒓 𝒏 ) 𝒏𝒕 𝑨=𝑷 𝒆 𝒓𝒕

Can you identify? Domain: Range: Y- intercept: Increase/decrease: Exponential Function Examples: Use your calculator to help you draw the function 𝑦 = 𝑒 𝑥 . Apply rules of transformations to draw 𝑦 = − 𝑒 𝑥−2 + 4 on the same graph, showing the reflections and translations in the graph. Can you identify? Domain: Range: Y- intercept: Increase/decrease: Asymptotes:

2. A total of $12,000 is invested at an annual interest rate of 9% 2. A total of $12,000 is invested at an annual interest rate of 9%. Find the balance after five years if it is compounded. Quarterly (n = 4). Monthly (n = 12) Continuously (𝐴 = 𝑃 𝑒 𝑟𝑡 ).