objectives 1.) to review and solidify basic exponential expressions and equations for the purpose of...
TRANSCRIPT
Objectives
1.) To review and solidify basic exponential expressions and equations for the purpose of further use in more complex exponential problems
Vocabulary
• A power is a number resulting from a number brought to an exponent.
• The parts of a power: Include a base number and an exponent.
• The base is based, while the exponent floats
5 3 = 125
Warm- upSolve the following perfect square problems:
12 = 92 =22 = 102 =32 = 112 =42 = 122 =52 = 132 =62 =72 =82 =
Quick Study TimeYour skills on perfect squares, cubes, powers of 2 and
powers of 3 will be tested.Cubed Powers:13 =1 23 = 8 33 = 2743 = 64 53 = 125
Base 2 Powers21 = 2 22 = 4 23 = 824 = 16 25 = 32 26 = 64
Base 3 Powers31 = 3 32 = 9 33 = 2734 = 81
Definition of Exponential Equations
Exponential functions are equations involving constants with exponents
Notated: y = ax
a= base; a>0 and not equal to 1 x = exponent/ power
Properties of exponents
n
b
a
a n 1
an
anm anm
1.) a0 = 2.) aman =
3.) (ab)m = 4.) (an)m =
5.) 6.)
7.)
1 nma
mmba mna
n
n
b
a
In-depth Look of Property # 6Negative exponents
Cross the line, flip the sign.
In- depth Look of Property #7Radicals versus Rational Exponents
x12 x
x13 x3
x14 x4
x15 x5
x16 x6
...
Can you solve the expression
with your calculator?
40964
Putting it all together
a
b
n
an
bn
a n 1
an
anm anm
3.) (ab)m = amam 4.) (an)m = anm
1.) a0 = 1 2.) aman = am+n
1
x
4
3
Write the expression using positive rational exponent
5.) 6.)
7.)
Graphs of Exponential Functions
Pg. 200
Graphs of exponential functions
x
f(x)f(x) = ax , a>1
x
f(x)f(x) = ax , a<1f(x) = a-x
Characteristics of Exponential Function Graphs
Transformations
Compound interest
nt
n
rIP
1
One lucky day , you find $8,000 on the street. At the Bank of Baker- that’s my bank, I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank.
After the first year, your account collects 10% interest, so I would have to payout 8000+8000(.1)= $8,800
Or, 8,000(1 + .1) = $8,800
The second year, your $8,800 will collect even more interest and become
8,800(1 + .1) = 8,000(1 +.1)(1+.1)= $9,680
Complete the table below
Year 1 2 3 4 5Payout Amou
nt
8,800 9,680 10,648 11, 71212,884
One lucky day , you find $8,000 on the street. At the Bank of Baker- that’s my bank. I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank.
Deal or No Deal?
You come to me with $5000. I have an interest rate of 4.1 %. You want to establish this amount in my bank for 20 years.
What if I compound your investment quarterly. I will apply a compounded interest rate 4 times but I will divide the interest rate by 4.
trIP 1
20041.15000 P
20*4
4
041.15000
P
Initial investment
Interest rate in decimal form
I will pay 4 times per year for 20 years, but as consequence I will divide interest rate by 4
11,168.24
11,305.21
Compound interest
nt
n n
rI
1lim
In 1683, mathematician Jacob Bernoulli considered the value of
as n approaches infinity. His study was the first approximation of e
n
n
11
e= 2.718281828459045235460287471352662497757246093699959574077078727723076630353547594571382178525166427466391932003059921817413496629043572900338298807531952510190115728241879307…..
Comparable to an irrational number like ∏
\