Exponential Functions and Their Graphs:
Algebraic functions are the functions that can be expressed by using the arithmetic operations and whose values are either rational or the root of rational number. Now, we will be dealing with transcendental functions. The Transcendental functions return values that might not be expressible as the rational numbers or roots of rational numbers.
The algebraic equations can be solved most of the time by hand. The transcendental functions can frequently be solved by hand with a calculator essential if you wish a decimal approximation. Though whenever transcendental and algebraic functions are mixed in an equation, graphical or numerical methods are sometimes the only way to find out the solution.
Simplest exponential function is as: f(x) = ax, a>0, a≠1
The main reasons for the restrictions are simple. When a≤0, then whenever you increase it to a rational power, you might not obtain a real number. Illustration: When a = -2, then (-2)0.5 = sqrt(-2) that is not real. When a = 1, then no matter what x is, the value of f(x) is 1. That is a quite boring function, and it is surely not one-to-one.
Remember that one-to-one functions had some properties which make them desirable. They contain inverses which are also functions. They can be applied to both sides of the equation.
Graphs of Exponential Functions:
The graph of y = 2x is as shown below. Here are a few properties of the exponential function whenever the base is more than 1.
a) The graph passes via the point (0,1).
b) The domain is all the real numbers.
c) The range is y > 0.
d) The graph is rising.
e) Graph is asymptotic to the x-axis as x approaches to the negative infinity.
f) The graph rises devoid of bound as x approaches to the positive infinity.
g) The graph is smooth.
h) Graph is continuous.
What would be the translation if you substituted each and every x with -x? This would be the reflection regarding y-axis. We as well know that whenever we increase a base to the negative power, the one outcome is that the reciprocal of number is taken. Therefore, if we were to graph y = 2-x, the graph would be a reflection regarding y-axis of y = 2x and the function would be equal to y = (1/2)x.
The graph of y=2-x is shown below. The properties of exponential function and its graph whenever the base is between 0 and 1 are specified.
d) The graph is reducing.
e) Graph is asymptotic to the x-axis as x approaches to the positive infinity.
f) The graph rises without bound as x approaches to the negative infinity.
g) The graph is continuous.
h) Graph is smooth.
Note that the only differences regard whether the function is decreasing or increasing, and the behavior at the left hand and right hand ends.Translations of Exponential Graphs:
You can apply what you know regarding translations to help you sketch the graph of exponential functions.
The horizontal translation might affect the increasing or decreasing (when multiplied by a negative), the left hand/right hand behavior of the graph, and the y-intercept, however it won't modify the place of the horizontal asymptote.
The vertical translation might influence the increasing or decreasing (when multiplied by a negative), the y-intercept, and the place of horizontal asymptote. This will not modify whether the graph goes with no bound or is asymptotic (though it might modify where it is asymptotic) to the right or left.
By knowing the characteristics of the fundamental graphs, you can apply such translations to simply sketch the new function.
You must now add up the exponential graph from the front cover of text to the list of such you know.
The natural base e:
Since x rises devoid of bound, the quantity (1+1/x)^x will approach to the transcendental number e.
The limit notation shown above is from calculus. The limit notation is a manner of asking what occurs to the expression as x approaches to the value shown. The limit is dividing line between calculus and algebra. The Calculus is algebra with the concept of limit. People always contain this dread of calculus. The calculus itself is simple. The reason people do not do well in calculus is not since of the calculus, however because they are poor at algebra.
The value for e is around 2.718281828. Here is a slightly more precise, although no more helpful, approximation.
2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45716 82178 52516 64274 27466 39193 20030 59921 81741 35966 29043 57290 03342 95260 59563 07381 32328 62794 34907 63233 82988 07531 95251 01901 15738 34187 93070 21540 89149 93488 41675 09244 76146 06680 82264 80016 84774 11853 74234 54424 37107 53907 77449 92069 55170 27618 38606 26133
Whenever the base e is used, the exponential function becomes f(x) = ex. There is a key on your calculator labeled e^x. On TI-8x calculators, it is on left side as a [2nd] [Ln]. The exponential function with base e is many times abbreviated as exp(). One common place this abbreviation emerges is whenever writing computer programs.
The amount in your saving account can be figured out with exponential functions. Each and every period (I'll suppose monthly), you obtain 1/12 of annual interest rate (r) applied to your account. New amount in the account is 100% of what you started with plus r%/12 of what you started with that. That signifies that you now contain (100%+r%/12) of what you started with. The later month, you will have similar thing, apart from it will be based on what you had at the end of first month.
It is little bit confusing. The resultant formula for compound interest is A = P (1+i)n.
A is the Amount in account. P is the principal you began with. i is the periodic rate, that is the annual percent (written as decimal) r, divided by the number of periods per year, m. n is the number of compounding periods, that is equal to the number of periods per year, m, times the time in years, t. The formula shown above varies slightly from the formula in books, however agrees with the formula that you will use when you go on to Finite Mathematics. In Finite Mathematics, there is a whole chapter on finance and the formulas included.
Continuous Compounding and Growth/Decay:
In old days, continuous compounding of interest is used. You do not find it anymore since it is gives the maximum return on the investment, and banks are in business to make money, just similar to any other for gain institution.
The model for continuous compounding is A = P ert.
Here, A is the Amount, P is the Principal, r is annual percentage rate (written as decimal), t is the time in years and e is the base for the natural logarithms.
Though, the continuous model does make sense for the population growth and radioactive decay. The radioactivity of isotope does not change once a month at the end of month, it is continually modifying.
The exponential model is y = A ekt,
Here, y is the amount present at time t. A is the initial amount present and k is the rate of growth (when positive) or the rate of decay (when negative).
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