Vocab

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Graphing Exponentials

Graphing Exponentials

Graphing exponential functions is similar to the graphing you have done before. However, by the nature of exponential functions, their points tend either to be very close to one fixed value or else to be too large to be conveniently graphed. There will generally be only a few points that are "reasonable" to use for drawing your picture; picking these sensible points will require that you have a good grasp of the general behavior of an exponential, so you can "fill in the gaps", so to speak.

Remember that the basic property of exponentials is that they change by a given proportion over a set interval. For instance, a medical isotope that decays to half the previous amount every twenty minutes and a bacteria culture that triples every day each exhibits exponential behavior, because, in a given set amount of time (twenty minutes and one day, respectively), the quantity has changed by a constant proportion (one-half as much and three times as much, respectively).

You can see this behavior in any basic exponential function, so we'll use y = 2x as representative of the entire class of functions:

On the left-hand side of the x-axis, the graph appears to be on the x-axis. But the x-axis represents y = 0. Can you ever turn "2" into "0" by raising it to a power? Of course not. And a positive "2" cannot turn into a negative number by raising it to a power, so the line, despite its appearance, never goes below the x-axis into negative y-values; the graph of y = 2^x is always actually above the x-axis, even if only by a vanishingly-small amount.


So why does it look like it is right on the axis? Remember what negative exponents do: they tell you to flip the base to the other side of the fraction line. So if x = –4, the exponential function above would give us 2^–4, which is 2⁴ = 16 and then flipped underneath to be ¹/₁₆, which is fairly small. By nature of exponentials, every time we go back (to the left) by 1 on the x-axis, the line is only half as high above the x-axis as it had been for the previous x-value. That is, while y = ¹/₁₆ for x = –4, the line will be only half as high, at y = ¹/₃₂, for x = –5. So, while the line never actually touches or crosses the x-axis, it sure gets darned close! This is why, practically speaking, the left-hand side of a basic exponential tends to be drawn right along the axis. If you zoom in close enough on the graph, you will eventually be able to see that the graph is really above the x-axis, but it's close enough to make no difference, at least as far as graphing is concerned.

If you are using TABLE or some similar feature of your graphing calculator to find plot points for your graph, you should be aware that your calculator will return a y-value of "0" for strongly-negative x-values. Your calculator can carry only so many decimal places, and eventually it just gives up and says "Hey, zero is close enough":

Compound Interest

Compound Interest

Interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Compound interest can be thought of as “interest on interest,” and will make a deposit or loan grow at a faster rate than simple interest, which is interest calculated only on the principal amount. The rate at which compound interest accrues depends on the frequency of compounding; the higher the number of compounding periods, the greater the compound interest. Thus, the amount of compound interest accrued on $100 compounded at 10% annually will be lower than that on $100 compounded at 5% semi-annually over the same time period. Compound interest is also known as compounding.

The formula for calculating compound interest is:

Compound Interest = Total amount of Principal and Interest in future (or Future Value) less Principal amount at present (or Present Value)

= [P (1 + i)ⁿ] – P

= P [(1 + i)ⁿ – 1]

(Where P = Principal, i = nominal annual interest rate in percentage terms, and n = number of compounding periods.)

If the number of compounding periods is more than once a year, "i" and "n" must be adjusted accordingly. The "i" must be divided by the number of compounding periods per year, and "n" is the number of compounding periods per year times the loan or deposit’s maturity period in years.

For example:

• The compound interest on $10,000 compounded annually at 10% (i = 10%) for 10 years (n = 10) would be = $25,937.42 - $10,000 = $15,937.42
• The amount of compound interest on $10,000 compounded semi-annually at 5% (i = 5%) for 10 years (n = 20) would be = $26,532.98 - $10,000 = $16,532.98
• The amount of compound interest on $10,000 compounded monthly at 10% (i = 0.833%) for 10 years (n = 120) would be = $27,070.41 - $10,000 = $17,070.41

Vertices:
Constraints	Objective Function:
x ≥ 0 y ≥ 0 x + y ≤ 5

 

 

 

Vertices:
Constraints	Objective Function:
x ≥ 0 y ≥ 0 x + y ≤ 5

 

 

Statistics Test

Statistics Test

Statistics

                                                       Statistics Data

General Forms of a Sequence

General Forms of Sequence

 

 

 

Geometric Sequence- has a common ratio

ex. an = a1 x rn-1 -----> 3 , 6 , 12 , ___ ---> a4 = 3 x 23

a4 = 24

Arithmatic                      
an = a1 (n-1) d

*ex. 1, 3, 5, 7, 9    ( increasing 2)

Geometric
an = a1 x rn-1

*ex. -9, 3, -1. 1/3. -1/9 (multipying by -1/3)

subscripts

exponents

Charactoristics & Traits of a Graph

Charatoristics & Traits of a Graph

I. Domain- left and right traits of a graph

                                                   Range- up and down traits of a graph

    

     II. End Behaivor-

                Left: x approaches negative infinity   Up: y approaches positive infinity

                Righ: x approaches positive infinity   Down: y approaches negative infinity

   

     III. X / Y Intercept

                x intercept: where it crosses x-axis {ex. (3,0)}

                y intercept: where it crosses y-axis {ex. (0,3)}

     IV. Function- passes the Vertical line test.
         One to One- passes the Horizonal and Vertical line test.

     V. Symmetry-
               even- symmetric about the y-axis
               odd- rotational symmetry around the origin
               niether- none at all

     VI.  Max/Min
          MAX- graph opens down            Absolute- 1 single point
          min- graph opens up                    Local- more than one single point