Graph each function:

1.f(x) = -2x2 – 4x + 3

2.f(x) = -x3 + 4x2

1.

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ExponentialAnd Logarithmic Functions

Pre-Cal

LinearQuadraticAbsolute ValueRadicalPolynomialRational

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ExponentialLogarithmic

These are special function that model many real life situations.

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1. f(x) = 2x x = -3.12. f(x) = 2-x x = π3. f(x) = 0.6x x = 3/2

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0.117

0.113

0.465

An equation in the form f(x) = abx-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) = 2x

f(x) = 2x

ExponentialDecay

(decreasing)

ExponentialGrowth

(increasing)

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What change do you notice in the graph when “k” changes?

Vertical Shift The graphs of f(x) = abx-h + k are shifted vertically by k units.

f(x) = (½)x f(x) = (½)x + 1 f(x) = (½)x – 3

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What change do you notice in the graph when “h” changes?

Horizontal Shift The graphs of f(x) = abx-h + k are shifted horizontally by h units.

f(x) = (2)x f(x) = (2)x–3 f(x) = (2)x+2 – 3

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What change do you notice in the graph when “a” changes?

The graphs of f(x) = abx-h + k are reflected, stretched and shrunk.

f(x) = (2)x f(x) = -(2)x f(x) = 3(2)x

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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 = -3x–1

b = 3 increasing

y-int:y = -30-1

y = -3-1

(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)0-3+1

y = 3(½)-3 + 1 (0, 25)

asy: y = k y = 1

shift: right 3, up 1

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3. y = 5x – 1

b = 5 increasing

y int:

y = 50-1 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+2-3 (0, -29/9)

asy: y = k y = -3

shift: left 2, down 3

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5. f(x)= (2/3)x

g(x) = -(⅔)x–2

Reflect over xShift right two

6. f(x) = (4/3)x

g(x) = (4/3)x–3 + 1

Right three

Up 1

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7. y = (½)-(x+1) – 3

Left one

Down three

Reflect over y

8. y = -(2.5)x+2 – 4

Reflect over xLeft twoDown four

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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”

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Graphy = ½(3)x

Asy: y = 0Base graph:

y= ½(3)x

No Shift

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x y

0

1

½ 3/2

Graphy = 5x – 3

Asy: y = -3Base graph:

y= (5)x

Shift down three

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x y

0

115

Graphy = 3(2)x+3

Asy: y = 0Base graph:

y= 3(2)x

Shift left three21

x y

0

13

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Graphy = -5(⅔)x-2 + 1

Asy: y = 1Base graph:

y= -5(2/3)x

Shift right two up one

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x y

0

1-5-31/3

Graphy = 5(½)-x

Asy: y = 0Base graph:

y= 5(½)-x

y = 5(2)x

No Shift23

x y

0

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Natural Base eEuler's NumberIt is a special number like π or i.

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

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1. f(x) = ex x = -22. f(x) = ex x = .253. f(x) = ex x = -0.4

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0.135

1.284

0.670

Standard form: f(x) = aer(x – h) + k If a > 0...

And r > 0, then the graph will be a growth▪ f(x) = 3e2(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)

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1. Graph base graph using table with x-values 0 and 1

2. Shift points

3. Graph asymptote y = k

4. Draw “smooth curve”

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Graph and state

domain and range.

y = 2e0.75x

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Graph and state domain and range.

y = e-0.5(x-2) + 1

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Graph and state domain and range.

y = e0.4(x+1) – 2

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Graph and state domain and range.

y = -3e0.5x

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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

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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

1993.

a)About how many hosts were there in 1996?

b)Graph using a calculator.

c)Estimate the year when there will be 30 million hosts.

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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 1990.

a)How much would tuition be in 1996?

b)Graph using a calculator.

c)Find out when tuition will be approximately $6000.

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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

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You buy a car for $24000. The value decreases 16% each year.

a)Write a decay model for the value of the car.

b)Estimate the value of the car after 2 years.

c)Use a calculator to graph.

d)Find out how long it would take for the car's value to reach $12,000.

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An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person’s system decreases

by about 29%.

a)Write an exponential model to represent the amount of ibuprofen left in the system after t hours.

b)How much ibuprofen is left after 6 hours?  

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During normal breathing, about 12% of the air in the lungs is replaced after one breath.

a)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.

b)How much of the original air is present after 24 breaths?

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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

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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?

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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.

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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.

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Since 1972 the US Fish and Wildlife Services haskept a list of endangered species in the US. For the years 1972–1998 the

number s of specieson the list can be modeled by

s = 119.6e0.0917t.

a)Find the number of endangered species in 1972.

b)Find the number in 1984.

c)Graph the model.

d)Use the graph to estimate when the number of endangered species reached 1000.

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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.

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You deposit $1000 in an account that pays 8% annual interest compounded continuously. What

is the balance after 1 year? After 3 years?

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Graph each:

1.y = -(3)x-2 + 1

2.y = 2(½)x – 3

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