Exploring Exponential Growth and Decay Activity -Take out warm-up -Take out a blank piece of paper

Objective – To be able to graph exponential growth and decay functions and to become familiar with asymptotes. State Standard – 12.0 Students understand exponential functions and use them in problems with exponential growth and decay. Asymptote – is the line that a graph approaches as you move away from the origin.

Exponential Functions – involves the expression y = ab x base – is the value b, with the exponent Exponential Growth If a > 0 and b > 1 –1 –5–4–3–2– y = 4(3) x Exponential Decay If a > 0 and 0 < b < 1 –1 –5–4–3–2– y = 3(½) x

Tell me if the equation is Growth, Decay, or Neither. a)y = 4( 1 / 4 ) x b)y = -3( 2 ) x c)y = 5( 3 / 7 ) x d)y = 8( 8 / 3 ) x Decay Neither Decay Growth

Example 1 Graph the function a) y = 2 / 3 (2) x 1) First draw an x/y box and find 2 points: –1 –5–4–3–2– x y /32/3 4/34/3

Example 2 Graph the function a)y = 4 ( 2 / 5 ) x 1) First draw an x/y box and find 2 points: –1 –5–4–3–2– x y /58/5

On White Board Graph the function a)y = 3 x 1) First draw an x/y box and find 2 points: –1 –5–4–3–2– x y

On White Board Graph the function a)y = 4( ½) x 1) First draw an x/y box and find 2 points: –1 –5–4–3–2– x y

Appreciation/Depreciation Model A = P(1 ± r) t Where A = Final Amount P = Initial Amount r = Percent Rate t = Time (usually in years)

Example 6 You buy a car for $4000 when you are 16 and find out it depreciates by 12 percent each year. How much is your car worth when you want to sell it when you are 20? First Identify your P, r, t, and if we are adding or subtracting. A = 4000 ( 1 – 0.12) 4 A = 4000 (0.88) 4 A = 4000(0.5997) A = So you could roughly sell your car for $2400 when you are 20.

Example 7 Now you buy a classic car for $4000 that appreciates in value 12% each year. How much is your car worth when you want to sell it when you are 20 A = 4000 ( ) 4 A = 4000 (1.12) 4 A = 4000( ) A = So you could roughly sell your car for $6300 when you are 20.

See Assignment Sheet

Simplify 1) (1 + ½) 2 2) (1 + 1 / 3 ) 3 3) (1 + 1 / 4 ) 4 ( 3 / 2 ) 2 9/4 9/4 ( 5 / 4 ) 4 ( 4 / 3 ) 3 64 / / 256

Objective – To be able to identify the role of “h” and “k” for exponential growth and decay. And use the “e” as a base. State Standard – 12.0 Students understand exponential functions and use them in problems with exponential growth and decay.

Example 1 Graph y = 23 (x-2) + 1 To graph: y = ab (x-h) + k, begin by graphing y = ab x. Then translate the graph horizontally by h units and vertically k units. 1) First graph y = 23 x x y –1 –5–4–3–2– ) Then move h units horizontally and k units vertically h = 2 k = 1

Example 2 Graph y = 5( 1 / 8 ) (x+2) – 2 To graph: y = ab (x-h) + k, begin by graphing y = ab x. Then translate the graph horizontally by h units and vertically k units. 1) First graph y = 5 ( 1 / 8 ) x x y /85/8 2) Then move h units horizontally and k units vertically h = -2 k = -2 –5–4–3–2– –5 –4 –3 –2 –

THE NATURAL BASE e The natural base e is irrational. It is defined as follows: As n approaches + , (1 + 1 / n ) n approaches e 

Example 3 Simplify the expression a)e 2 e 3 b) (20e 4 )/(5e 3 ) c) (4e -3x ) 2 e 2+3 e5e5 4e 4-3 4e 1 16e (-3x)2 16e (-6x) 16 e 6x 4e

Example 4 Graph the function a)y = 3e 0.5x b) y = e ( 0.4x+1) – 2 1)First draw an x/y box and find 2 points: –1 –5–4–3–2– x y x y –5–4–3–2– –5 –4 –3 –2 – ) First graph y = e 0.4x h = -1 k = -2

Find the value of x. 1) 3 x = 9 2) x 3 = -8 3) 10 0 = x4) ( 3 / 2 ) -1 = x /32/3

The Richter Scale Magnitude E E(30) E(30) 2 E(30) 3 E(30) 4 E(30) 5 E(30) 6 E(30) 7 E(30) 8 E(30) 9 energy released: x 30

8.3 Logarithmic Functions Objective – To be able to evaluate and graph logarithmic functions. State Standard – 11.1 Students solve problems involving logarithms and exponents DEFINITION OF LOGARITHM WITH BASE b log b y = x if and only if b x = y The expression log b y is read as “log base b of y.”

Example 1 Rewrite the logarithmic equation in exponential form. a)log 3 9 = 2 b) log 8 1 = 0 c) log 5 ( 1 / 25 ) = = = 15 (-2) = 1 / 25 Example 2 Evaluate the expression. a)log 4 64 b) log 3 27 c) log 6 ( 1 / 36 ) 4 x = 64 x = 3 3 x = 276 x = 1 / 36 x = 3 x = -2

Common Logarithm Common Logarithm – Is a log with base 10. ie. log 10 or simply just log so…… log 10 x = log x GRAPHS OF LOGARITHMIC FUNCTIONS y = log b (x – h) + k if 0 1 the graph moves up

Example 3 Graph the function y = log 3 (x – 2) 1)First draw an x/y box and find 2 points for the graph y = log 3 x x y –5–4–3–2– –5 –4 –3 –2 – h = 2 k = 0 3 y = x

On White Board Graph the function y = log 5 (x + 1) + 2 1)First draw an x/y box and find 2 points for the graph y = log 5 x x y –5–4–3–2– –5 –4 –3 –2 – h = -1 k = 2 5 y = x

With what we learned from 8.3, simplify 1) log log 10,000 2) log log ) log – log ) 2 log 5 25 (10 x = 100) and (10 x = 10,000) 3 – 2 = 1 (5 x = 25) and (5 x =125) 2 2 = 4 x = 2 and x = = 6 x = 2 and x = = 5

Objective – To be able to use properties of logarithms State Standard – 11.0 Students will understand and use simple laws of logarithms. Properties of Logarithms Product Propertylog b uv = log b u + log b v Quotient Propertylog b (u/v) = log b u - log b v Power Propertylog b u n = n log b u

Example 1 State the property used: a. log 2 + log 3 = log 6 b. log 3 (x/y) = log 3 x – log 3 y c. log 4 (x 2 y) = 2log 4 x + log 4 y Product Property Quotient Property Product and Power Property

log log 5 x 6 Product Property log b uv = log b u + log b v Power Property log b u n = n log b u Example 2 Expand log 5 2x 6 Which property is being used? log log 5 x

Example 3 Condense: 2 log 3 7 – 5 log 3 x log – log 3 x 5 = log – log 3 x 5 log = log x 5 = log 3 49 x 5

On White Board Expand: log 5 3r 4 log log 5 r 4 = log log 5 r 4 log log 5 r = log log 5 r

On White Board Condense: log 4 12 – log 4 2 log 4 12 = log = log 4 6

1. Simplify /3 2. Evaluate log Evaluate log /

8.5 Solving Exponential and Logarithmic Equations Objective – Students will be able to solve exponential equations. CA State Standard – 11.0 Students use simple laws of logarithms.

Properties of Exponential and Logarithmic Functions 1. x = yif and only if a x = a y 2. x = y if and only if log a x = log a y 3. log a a x = x

Evaluate 2 4x = 32 x-1 4x = 5x – 5 Example x = 2 5(x-1) +5 4x + 5 = 5x 5 = x -4x

log c u = Change - of - Base Formula

Evaluate 4 x = 15 x = Example 2 log 4 4 x = log 4 15 x = 1.95 log 15 log 4 x = log 4 15

Evaluate log 4 (x+3) = log 4 (8x+17) Example 3 x + 3 = 8x + 17 x = -2 -x 3 = 7x = 7x