Exponential Functions

Concavity

100

determine the concavity of the function at x=0

f(x)=2^x

concave up

200

find the interval where the function is concave up or down

f(x)= e^3x

use inf for infinity

(-inf,inf) concave up

300

For the function find the interval for the concavity 

f(x)=5e^-x

use inf for infinity

concave up (-inf,inf)

400

For the function determine the concavity of the function 

f(x)=4e^2x

use inf for infinity

concave up (-inf,inf)

500

for the unction find the intervals of concavity and identify any points of inflection

f(x)=e^(x)-2e^(-x+4)

concave down (-inf,1/2ln(2))

concave up (1/2ln(2),inf)

point of inflect at x=1/2ln(2)

Inverse

100

What is the inverse of e

ln

200

Solve for x

3^x=27

x=log3(27)

Constant becomes the log base.

300

Solve for x:

log2(x)=5

x=32

The log base is raised to the power of the constant on the other side of the equation

400

f(x)=(e^x^3)+1 Find the Inverse

Inverse= 3sqrt(ln(x-1))

Switch x and y

Subtract 1 from both sides

Take the natural log

Take the cubed root from both sides

500

f(x)=3sqrt(7e^x). Find the inverse

Inverse= ln(x^3/7)

Switch x and y

Cube both sides

Divide both sides by 7

Take the natural log

Application

100

A town has a population of 12,000 people, and the population is growing at an annual rate of 3%.

• Write an exponential growth equation to model the population.
• Use the equation to find the population after 5 years.

13,911

200

A certain radioactive substance decays at a rate of 5% per year. If the initial amount of the substance is 100 grams:

-Write an exponential decay equation to model the substance over time.

-How much of the substance will remain after 10 years?

59.87

300

The number of bacteria in a culture doubles every 4 hours. Initially, there are 50 bacteria.

• Write an equation to represent the population of bacteria as a function of time in hours.
• How long will it take for the population to reach 1,600 bacteria?

20 hours

400

A certain investment account earns continuous interest at an annual rate of 6%. If $5,000 is invested:

• Write the equation for the amount of money in the account as a function of time, A(t)=Pe^rt
• How much money will be in the account after 8 years?

$8,085

500

A cup of coffee is initially 90°C and is left to cool in a room with a temperature of 20°C. The coffee cools according to Newton’s Law of Cooling:

T(t)=T-room+(T-initial−T-room)e^−kt

where T-room=20C, T-initial = 90C, and K is a cooling constant.

• If the coffee cools to 70°C in 10 minutes, determine the value of k.

0.0369

Transformations

100

Equation: 3e^x-2

Identify the transformations.

 Veritical stretch by 3 

Right by 2

200

Equation: 5e^(x+3) -3 

Identify the transformations.

Vertical Stretch by 5

Left by 3

Down 3

300

Equation:-3^(x+2)+5

Identify the transformations.

Reflection across X-axis

Left by 2

Up 5

400

The Fucntion F(x)=2^x is transformed into g(x)= -4(2)^(x-1) + 3. Identify the transformations applied to f(x). 

Vertical Stretch by 4

Refection across X-axis

Right by 1

Up by 3

500

Function f(x)=e^2(x) is transformed into g(x)=-3(e)^(x/2-1)+4. Identify the transformations applied to f(x).

Reflection over x-axis

Vertical Stretch by 3

Horizontal stretch by a factor of 2

Right 1

Up 4

Formula

100

What is the Exponential Equation (General Form)?

Let a = initial

Let b= base 

ab^x

200

What is the exponential growth/decay formula?

let I = initial 

let r = growth rate

let t = time

I(1+r)^t

300

What is the Half-Life formula?

Let I = initial

Let t = time

Let h = how long it takes for half of the substance to decay

I(1/2)^t/h

400

What is the Compound interest formula?

Let P = Principle

Let R = Rate

Let N = Number of time compounded

Let T = Time

P(1+r/n)^nt

500

What is the continuous compund interest formula?

Let P = Principle

Let R = Annual interest rate

Let T = Time

Pe^rt