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Exponential Functions • Exponential functions • Geometric Sequences

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Exponential and Logarithmic Functions

Exponential FunctionsExponential functionsGeometric Sequences8.1 Exponential functions and their graphsRecognize and evaluate exponential functions with base aGraph exponential functionsRecognize and evaluate exponential functions with base eUse exponential functions to model and solve real-life problems

Exponential functions

Graph:

To graph exponential functions, make a table, plot the points and connect them with a smooth curve. x f(x) g(x) -1 0 1 2Domain:Range:y -intercept:4Translations y = bx - original graph

y = abx-h + k

k - positive - moves graph upk - negative - moves graph down

h - positive - to the righth - negative - to the left

a < 0 - reflected over x - axisa > 1 - stretched0< a < 1 - compressed

Graph:

greenDomain:Range:y -interceptDomain:Range:y -intercept

A transformation of the graph:

x yDomain:Range:8.2 Solving exponential equations

To solve exponential equations, get the bases equal.

Solve for x:

One to one property!

Bases must be the same

BONUS!!Compound Interest FormulasAfter t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formula:1) For n compounding per year:

Find the account balance after 20 years if $100 is placed in an account that pays 1.2% interest compounded twice a month.If $350,000 is invested at a rate of 5% per year, find the amount of the investment at the end of 10 years for the following compounding methods:a) Quarterly b) MonthlyContinuously Compound InterestA = PertJoan was born and her parents deposited $2000 into a college savings account paying 4% interest compounded continuously. What would be the balance after 15 years.exponential decay:

P = initial amount(1 - r)/(1 + r) is the decay/growth factor, r is the decay rate0 < r < 1 t is the time periodA = final amountYou bought a used car for $18,000. The value of the car will be less each year because of depreciation. The car depreciates (loses value) at the rate of 12% per year. Write an exponential decay model to represent the situation then use that model to estimate the value of the car in 8 years.

exponential growth:

Graph and find the average rate of change in value from year 0 to year 4 t y

yearsvalue220008.3 Logarithmic FunctionsWrite exponential functions as logarithmsWrite logarithmic functions as exponential functions

A logarithm function is another way to write an exponential function

y is the logarithm, a is the base, x is the numberWe can now convert from one form to the other.

Rewrite as a exponential equation:

Rewrite as a logarithm:

To find the exact value of a logarithm (or evaluate), we can change the equation to an exponential one.

Evaluate:log3 81log1/2 256log13 169Evaluate:

11.3/11.4 Sum of Geometric Sequences

Nth term of geometric sequence: (this is used to find any of the items or terms) an = a1rn-1

a1 = 1st termr = common ratio (divide any term by the prior term)n = how many termsWrite an equation for the nth term of a geometric sequence -.25, 2, -16, 128, ...Find the equation for the geometric sequence: If a3 = 16, and r = 4

The geometric means are terms between non-consecutive terms of a geometric sequence.Find the 4 geometric means between .5, __, __, __, __, 512

Partial Sum of a Geometric SeriesFind the sum of the geometric seriesa1 = 2, n = 10, r = 3

Find the sum of the geometric seriesa1 = 2000, an = 125, r = 1/2

Find a1 in a geometric series for which Sn = -26240, n = 8, r = -3

Or use the sum formula:

Find the sum:

Find the sum: