Graph each function: 1. f(x) = -2x 2 – 4x f(x) = -x 3 + 4x

Exponential And Logarithmic Functions Pre-Cal

 Linear  Quadratic  Absolute Value  Radical  Polynomial  Rational 3

 Exponential  Logarithmic  These are special function that model many real life situations. 4

 f(x) = 2 x x = -3.1  f(x) = 2 -x x = π  f(x) = 0.6 x x = 3/

An equation in the form f(x) = ab x-h + k. if 0 < b < 1, the graph represents exponential decay if b > 1, the graph represents exponential growth Examples: f(x) = (½) x f(x) = 2 x f(x) = 2 x Exponential Decay (decreasing) Exponential Growth (increasing) 6

What change do you notice in the graph when “k” changes? Vertical Shift  The graphs of f(x) = ab x-h + k are shifted vertically by k units. f(x) = (½) x f(x) = (½) x + 1f(x) = (½) x – 3 7

What change do you notice in the graph when “h” changes? Horizontal Shift  The graphs of f(x) = ab x-h + k are shifted horizontally by h units. f(x) = (2) x f(x) = (2) x–3 f(x) = (2) x+2 – 3 8

What change do you notice in the graph when “a” changes? The graphs of f(x) = ab x-h + k are reflected, stretched and shrunk. f(x) = (2) x f(x) = -(2) x f(x) = 3(2) x 9

 An asymptote is a line that a graph approaches but never touches  The equation for the asymptote of an exponential function is y = k.  The y-intercept is a point where a graph crosses the y-axis.  plug in 0 for x to find the y-intercept  The y-intercept for the base graph is (0, a). 10

1. y = -3 x–1 b = 3  increasing y-int: y = y = (0, -1/3) asy: y = k y = 0 shift: right 1 2. y = 3(½) x–3 + 1 b = ½  decreasing y-int: y = 3(½) y = 3(½) (0, 25) asy: y = k y = 1 shift: right 3, up 1 11

3. y = 5 x – 1 b = 5  increasing y int: y = y = 0 (0, 0) asy: y = k y = -1 shift: down one 4. y = -2(⅓) x+2 – 3 b = ⅓  decreasing y int: y = -2(⅓) (0, -29/9) asy: y = k y = -3 shift: left 2, down 3 12

5. f(x)= ( 2 / 3 ) x g(x) = -(⅔) x–2 Reflect over x Shift right two 6. f(x) = (4/3) x g(x) = (4/3) x–3 + 1 Right three Up 1 13

7. y = (½) -(x+1) – 3 Left one Down three Reflect over y 8. y = -(2.5) x+2 – 4 Reflect over x Left two Down four 14

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1. Graph base graph using table with x-values 0 and 1 2. Graph asymptote  y = k 3. Shift points 4. Draw “smooth curve” 18

Graph y = ½(3) x Asy: y = 0 Base graph: y= ½(3) x No Shift 19 x y 0 1 ½ 3/23/2

Graph y = 5 x – 3 Asy: y = -3 Base graph: y= (5) x Shift down three 20 xy

Graph y = 3(2) x+3 Asy: y = 0 Base graph: y= 3(2) x Shift left three 21 xy

Graph y = -5(⅔) x Asy: y = 1 Base graph: y= -5( 2 / 3 ) x Shift right two up one 22 xy / 3

Graph y = 5(½) -x Asy: y = 0 Base graph: y= 5(½) -x y = 5(2) x No Shift 23 xy

 Natural Base e  Euler's Number  It is a special number like π or i. 24

 Go to table and see what happens as x gets larger… 25

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 f(x) = e x x = -2  f(x) = e x x =.25  f(x) = e x x =

 Standard form: f(x) = ae r(x – h) + k  If a > 0...  And r > 0, then the graph will be a growth ▪f(x) = 3e 2(x + 1) – 2  And r < 0, then the graph will be a decay ▪f(x) = 2e -3x + 5  The asymptote is y = k.  The y intercept of the base graph is (0, a) 28

1. Graph base graph using table with x-values 0 and 1 2. Shift points 3. Graph asymptote  y = k 4. Draw “smooth curve” 29

Graph and state domain and range. y = 2e 0.75x 30

Graph and state domain and range. y = e -0.5(x-2)

Graph and state domain and range. y = e 0.4(x+1) – 2 32

Graph and state domain and range. y = -3e 0.5x 33

y = a(1 + r) t y = amount after t years a = initial amount r = percent increase as decimal (1 + r) is the growth factor t = time in years 34

In 1993, there were 1,313,000 internet hosts. During the next 5 years, the number of hosts increased by 100% per year. Write a model giving the number h, hosts in millions, letting t represent the number of years since  About how many hosts were there in 1996?  Graph using a calculator.  Estimate the year when there will be 30 million hosts. 35

In 1990, the cost of tuition at a state University was $4300. During the next 8 years the tuition rose 4% each year. Write a model that gives tuition y in dollars and t, the number of years since a) How much would tuition be in 1996? b) Graph using a calculator. c) Find out when tuition will be approximately $

y = a(1 – r) t y = amount after t years a = initial amount r = percent decrease as decimal (1 – r) is the decay factor t = time in years 37

You buy a car for $ The value decreases 16% each year.  Write a decay model for the value of the car.  Estimate the value of the car after 2 years.  Use a calculator to graph.  Find out how long it would take for the car's value to reach $12,

An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person’s system decreases by about 29%.  Write an exponential model to represent the amount of ibuprofen left in the system after t hours.  How much ibuprofen is left after 6 hours? 39

During normal breathing, about 12% of the air in the lungs is replaced after one breath.  Write an exponential decay model for the amount of the original air left in the lungs if the initial amount of air in the lungs is 500 mL.  How much of the original air is present after 24 breaths? 40

A = p(1 + r/n) nt A = Final amount p = principal (beginning amount) r = % as decimal n = number of times compounded t = amount of time in years 41

You deposit $1000 in an account that pays 8% annual interest. Find the balance after one year if the interest is compounded quarterly. If compounded monthly? 42

You deposit $1600 in a bank account. Find the balance after 3 years for each of the following situations: The account pays 2.5% annual interest compounded monthly. The account pays 1.75% annual interest compounded quarterly. The account pays 4% annual interest compounded yearly. 43

If you invested $1,000 in an account paying an annual percentage rate (quoted rate) of 12%, compound quarterly, how much would you have in you account at the end of 1 year, 10 years, 20 years, 100 years. 44

Since 1972 the US Fish and Wildlife Services has kept a list of endangered species in the US. For the years 1972–1998 the number s of species on the list can be modeled by s = 119.6e t. a) Find the number of endangered species in b) Find the number in c) Graph the model. d) Use the graph to estimate when the number of endangered species reached

Recall the formula for compounding interest: This formula is for compounding interest a finite number of times. When interest is compounded continually, we use the formula where A is still the end amount, P the principal, r the interest rate, and t the time in years. 46

You deposit $1000 in an account that pays 8% annual interest compounded continuously. What is the balance after 1 year? After 3 years? 47

Graph each:  y = -(3) x  y = 2(½) x – 3 48