Math 170 4.1-4.6

An initial investment of 100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?

$3,644,675.88

How to graph an exponential function in the form: F(x)=b^x 1. Create a table of points: (__,__), (__,__), (___,___). 2.Plot points from table 3.Draw curve through points 4.State domain, range, and HA

(-1,1/b), (0,1), (1,b)

Find the domain of the following logs: f(x)=log3(x-2)+1

(2, infinity)

If s, t, a, and b are real numbers with a>0 and b>0 then, (a/b)^s=?

(a^s)/(b^s)

If s, t, a, and b are real numbers with a>0 and b>0 then, (a/b)^-s=?

(b/a)^s

Find the value of the expression without using a calculator: Log5(1/5)

-1

Graphing transformations of exponential functions: Form: f(x)=b^(x)+d Translation: ___________ Form: f(x)=^(x+c) Translation:___________ Form: f(x)=a(b)^x Translation:_________ Form: f(x)=-(b^x) Translation: Form: f(x)=b^-x=(1/b)^x Translation:

-Shift vertically d units, in the same direction of the sign d -Shift horizontally c units, in the opposite direction of the sign c -Stretch vertically by a factor of a if |a|>1. Compress vertically by a factor of a if 0<|a|<1 -Reflect about the x-axis -Reflect about the y-axis

Characteristics of a parent exponential function: f(x)=b^x -One-to-one function -Horizontal asymptote at y=___ -Domain: ________ -Range:______ -x-int: ___ -y-int:(__,__) -Increasing if b (>/<) 1 -Decreasing if b (>/<) 1

0 (-infinity, infinity) (0,infinity) none (0,1) > <

If s, t, a, and b are real numbers with a>0 and >0 then, a^0=?

1

Is s, t, a, and b are real numbers with a>0 and b>0 then, 1^s=?

1

Evaluate the base 7 log expression without using a calculator: Log7(square root of 7)

1/2

If s, t, a, and b are real numbers with a>0 and b>0 then, a^-s=?

1/a^s or (1/a)^s

Let f(x)=5(3)^(x+1). Evaluate the following without using a calculator: f(2) f(-1) f(-4) f(3)

135 5 5/27 405

(Using a calculator) Solve 42=1.2(5)^x+2.8 graphically. Round to the nearest thousandth.

2.166

Find an exponential function using a calculator that passes through the points (-2,6) and (2,1).

2.45(0.64)^x

Compound interest can be calculated using the formula: ?

A(t)=P(1+(r/n))^(nt)

The ___________ ____________/__________ Formula For all real numbers t, and all positive numbers a and r, continuous growth or decay is represented by the formula: A(t)=ae^(rt) Where, a is the _________ ___________ r is the __________ ___________ __________ per unit t is ___________ ___________ if r>0, then the formula represents continuous ______. If r<0, the the formula represents continuous __________.

Continuous Growth/ Decay initial value continuous growth rate elapsed time growth decay

A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013, the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population of wolves, N, over time, t.

N(t)=129(1.35)^t

In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population was growing exponentially. Write an exponential function N(t) representing the population of deer, N, over time.

N(t)=80(1.14)^t

For any real number x and any positive numbers a and b such that b is not equal to 1, and exponential growth function has the form: F(x)=ab^x Where, ___ is the initial or starting value of the function ___ is the growth factor of growth multiplier per unit ___.

a, b, x

Find the domain of the following logs: a.f(x)=log3(x-2)+1 b.g(x)=log1/2(x^2) c.h(x)=log2((x+3)/(x-1))

a. (2,infinity) b.(-infinity,0)U(0,infinity) c.(-infinity,-3)U(1,infinity)

Write the following log equations in exponential form: a. Log 6(square root of 6)=1/2 b. log5(25)=2 c. logz(12)=6

a. 6^1/2= square root root of 6 b.5^2=25 c.z^6=12

Write the following log equations in exponential form: a.3^2=9 b.x^-2=12 c.e^x=10

a.log3(9)=2 b.logx(12)=-2 c.loge(10)x

If s, t, a, and b are real numbers with a >0 and b>0 then, a^s x a^t = ?

a^(s+t)

If s, t, a, and b are real numbers with a>0 and b>0 then, (1/a)^-s=?

a^s

If s, t, a, and b are real numbers with a>0 and b>0 then, (ab)^s=?

a^s x b^s

If s, t, a, and b are real numbers with a>0 and b>0 then, (a^s)^t=?

a^st

Compound interest can be calculated using the formula: A(t)=P(1+(r/n))^(nt) Where, A(t) is the _____ ______ t is ____ in _______ P is the ______ _______ of the ____ r is n is the number of ________ __________ in one year.

account value time years starting amount account the annual percentage rate compounding periods

A function that models exponential growth, grows by a rate proportional to the _______ ________.

amount present

Which of the following represents an exponential function? a. f(x)=2x^2-3x+1 b. f(x)=0.875^x c. f(x)=1.75x+2 d. f(x)=1095.6^(-2x)

b, d

Graphing transformations of exponential functions: Horizontal transformations will occur in the _________ of the exponential function and vertical transformations will occur _________ of the _______.

exponent outside exponent

Write an equation for the function described below. Give the HA, D, and R. f(x)=e^x is vertically stretched by a factor of 2, reflected across the y-axis, and then shifted up 4 units.

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

Given these two points, find an equation for the exponential function: (0,3)(2,12)

f(x)=3(2)^x

For any real number x, an exponential function is a function with the form _______.

f(x)=ab^x

Exponential functions and their logs are inverses, so the inverse of f(x)=2^x is_______________.

g(x)=log2(x)

For business applications, the continuous growth formula is called compounding formula and takes the form: A(t)=Pe^(rt) Where, P is the ___________ ___________ r is the ________ _______ per unit time t is the ___________ of ______ of the investment

initial investment interest rate period term

What role does a horizontal asymptote of an exponential function play in telling us about the end behavior of the graph? -The horizontal asymptote tells us the ___________ of the function's values as the __________ variable gets either extremely large of extremely small.

limit independent

What is the vertical asymptote of f(x)=-2log3(x+4)+5?

x=-4

using a graphing calculator to approximate the solution of an equation: Solve 42=1.2(5)^8+2.8 graphically. Round to the nearest thousandth.

x=2.166

Solve each of the following without using a calculator: a.y=log121(11)

y=11

Evaluate y=log(321) to four decimal places using a calculator

y=2.5065

Evaluate y=log(1000) without using a calculator:

y=3

Evaluate the following to 4 decimal places using a calculator: y=ln(500) ln(-500)

y=6.2146 UDF