Warm Up 12/1 A special window in the shape of a rectangle with semicircles at each end is to be constructed so that the outside dimensions are 10 ft. in length. See picture. Find the dimensions of the rectangle that maximizes its area of the window. Hint: Circumference of a circle = 2pr 10 = 2l + 2pr (s0lve for l) Maximize Area of RECTANGLE: A = lw W = 2r Answer: w = 5/p l = 2.5

11.3 Exponential Functions SWBAT Evaluate and graph exponential functions. Define the number e

Laws of Exponents: If s, t, a and b are real numbers, with a >0 and b >0, 𝒂 𝒔 ∙ 𝒂 𝒕 = 𝒂 𝒔+𝒕 ( 𝒂 𝒔 ) 𝒕 = 𝒂 𝒔∙𝒕 (𝑎𝑏) 𝑠 = 𝑎 𝑠 ∙ 𝑏 𝑠 1 𝑠 =1 𝑎 −𝑠 = 1 𝑎 𝑠 = ( 1 𝑎 ) 𝑠 𝑎 0 =1 An Exponential Function is of the form: 𝑓 𝑥 = 𝑎 𝑥 Where “a” is positive. The domain of “f” is the set of all reals.

Use a calculator to evaluate each and round to four decimal places: Ex. 𝟑 𝟑 Ex. 𝒆 −𝟏.𝟖𝟓 ≈6.7050 ≈.1572

From the table decide if each function is exponential ( From the table decide if each function is exponential (*Check to see if ratio of consecutive outputs is the same based on a 1 unit increase in the inputs.) If it is find “a”. x F(x) -1 2 5 1 8 11 x G(x) -1 1/2 1/4 1 1/8 2 1/16 1 4 1 2 = 1 8 1 4 = 1 2 = a Exponential 5 2 ≠ 8 5 Not Exponential An exponential function 𝑓 𝑥 = 𝑎 𝑥 , 𝑎>0, 𝑎≠1, 𝑖𝑓 𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑒𝑛 𝑓(𝑥+1) 𝑓(𝑥) =𝑎

Graphing an exponential function: Ending Behavior: As x approaches ∞, y approaches 0. As x approaches -∞, y approaches ∞.

Properties of Exponential Functions in the form 𝑓 𝑥 = 𝑎 𝑥 𝑎𝑛𝑑 𝑎>1: Domain is set of all Reals. Range is the set of all positive reals. No x-int. Y-int is (0,1). Why? The line y=0 is a horizontal asymptote and x becomes unbounded in negative direction. 𝑥→∞, 𝑦→∞ 𝑎𝑛𝑑 𝑥→−∞, 𝑦→0 Its an increasing function. Graph contains (0,1), (1,a), and (-1,1/a). Graph is continuous with no corners or gaps.

Graphing with shifts and reflections: 𝑓 𝑥 = 𝑎 𝑥−ℎ +𝑘 -h = shift right +h = shift left +k = shift up -k = shift down -x = reflect over y axis - Out front = reflect over x axis.

Graph: 𝑓 𝑥 =− 3 𝑥 +1 From original graph; reflect over x axis and shift up 1.

Solve each exponential: Make the bases the same and eliminate them: 4 𝑥 2 = 2 𝑥 2 2 𝑥 2 = 2 𝑥 2 𝑥 2 =𝑥 2 𝑥 2 −𝑥=0 𝑥 2𝑥−1 =0 𝑥=0, 𝑥= 1 2

HW: 11.3A #s: 1, 11-17odds, 19-26all, 27,29, 35-45 odds