What is the end behavior of a parabola,history of ed sheeran,communication skills meaning and importance - PDF 2016

admin | Category: Improving Erections | 20.10.2014
We learned what a Polynomial is here in the Introduction to Multiplying Polynomials section. As a review, here are some polynomials, their names, and their degrees.  Remember that the degree of the polynomial is the highest exponent of one of the terms (add exponents if there are more than one variable in that term).
We learned that a Quadratic Function is a special type of polynomial with degree 2; these have either a cup-up or cup-down shape, depending on whether the leading term (one with the biggest exponent) is positive or negative, respectively. There are certain rules for sketching polynomial functions, like we had for graphing rational functions. Again, the degree of a polynomial is the highest exponent if you look at all the terms (you may have to add exponents, if you have a factored form).   The leading coefficient of the polynomial is the number before the variable that has the highest exponent (the highest degree). So for, the degree is 4, and the leading coefficient is 5; for, the degree is 7 (add exponents since the polynomial isn’t multiplied out and don’t forget the x to the first power), and the leading coefficient is –10 (you can tell by the –5 in front and the 2x in the factor with the highest exponent).
Each factor in a polynomial has what we call a multiplicity, which just means how many times it’s multiplied by itself in the polynomial (its exponent). Remember that x – 4 is a factor, while 4 is a root (zero, solution, x-intercept, or value).
Now we can use the multiplicity of each factor to know what happens to the graph for that root – it tells us the shape of the graph at that root. Also note that you won’t be able to determine how low and high the curves are when you sketch the graph; you’ll just want to get the basic shape.
Here are a few problems where we use the Conjugate Zeroes Theorem and Complex Conjugate Zeroes Theorem (also called Conjugate Root Theorem or Conjugate Pair Theorem), which states that if    is a root, then so is  .  The complex form of this theorem states that if    is a root, then so is  .
Also remember that when we factor to solve quadratics or any polynomials, we can never just divide by factors (with variables) on both sides to get rid of them.  If we do this, we may be missing solutions! Many times we’re given a polynomial in Standard Form, and we need to find the zeros or roots. For higher level polynomials, the factoring can be a bit trickier, but it can be sort of fun — like a puzzle! Remember that if we divide a polynomial by “x – c” and get a remainder of 0,  then “x – c” is a factor of the polynomial and “c” is a root.
When we want to factor and get the roots of a higher degree polynomial using synthetic division, it can be difficult to know where to start!   In the examples so far, we’ve had a root to start, and then gone from there. For a polynomial function  with integers as coefficients (no fractions or decimals), if p = the factors of the constant (in our case, d), and q = the factors of the highest degree coefficient (in our case, a), then the possible rational zeros or roots are  where p are all the factors of d above, and q are all the factors of a above. Remember that factors are numbers that divide perfectly into the larger number; for example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The rational root test help us find initial roots to test with synthetic division,  or even by evaluating the polynomial to see if we get 0.  However, it doesn’t make a lot of sense to use this test unless there are just a few to try, like in the first case above.


Now let’s try to find roots of polynomial functions without having a first root to try. Also remember that you may end up with imaginary numbers as roots, like we did with quadratics. Take out any Greatest Common Factors (GCFs) of the polynomial, and you’ll have to set those to 0 too, to get any extra roots.  For example, if you take an x out, you’ll add a root of “0”. If you have access to a graphing calculator, graph the function and determine if there are any rational zeros with which you can use synthetic division.   If you don’t have a calculator, guess a possible rational zero using the  method above.
There are a couple of theorems that you’ll learn about that will help you evaluate polynomials (for a given x, find the y) and also be able to quickly tell if a given number is a root. There’s another really neat trick out there that you may not talk about in High School, but it’s good to talk about and pretty easy to understand.  Yes, and it was named after a French guy!
The DesCartes’ Rule of Signs will tell you the number of positive and negative real roots of a polynomial  by looking at the sign changes of the terms of that polynomial. We talked a little bit about the Complex Conjugate Zeros Theorem here when we talked about all the steps required to find all the factors and roots of a polynomial. When we study about functions and polynomial, we often come across the concept of end behavior. End behavior is useful to examine the trend in the function value as the value of x gets larger and larger in magnitude.
End behavior of a polynomial function is the behavior of the graph of y = f(x) as x approaches positive infinity or negative infinity. Now, we can find the end behavior by just knowing the values of Leading Co-efficient an and Power n of the Polynomial Equation. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph. Case 1: If Leading coefficient(an) is positive and the power(n) is even, both ends of the graph go up.
Case 2: If Leading coefficient(an) is negative and the power(n) is even, both ends of the graph go down. Case 3: If Leading coefficient(an) is positive and the power(n) is odd, the right hand of the graph goes up and the left hand goes down. Case 4: If Leading coefficient(an) is negative and the power(n) is odd, the right hand of the graph goes down and the left hand goes up. Asymptotic behavior of graph of a function involves limits, since limits are the situations where a function approaches a value. The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction.


An end behavior model of a polynomial uses only the leading coefficient and the variable of highest degree. End behavior of a graph can be based on the degree and the leading coefficient of a polynomial function.
It is clear from the graph that that when x approaches to $\infty$, the right part of the graph is going upward. From the graph, when x approaches to $\infty$, the right part of graph extends towards upward direction. End Behavior refers to the behavior of a graph as it approaches either negative infinity, or positive infinity.
Notice that the negative part of the graph is more of a “cup down” and the positive is more of a “cup up”. If there is no exponent for that factor, the multiplicity is 1 (which is actually its exponent!)  And remember that if you sum up all the multiplicities of the polynomial, you will get the degree!
As the name suggests, "end behavior" of a function is referred to the behavior or tendency of a function or polynomial when it reaches towards its extreme points. It helps to tell us how does f(x) behave as the value of x increases to positive infinity or decrease to negative infinity. In other words, we are interested in what is happening to the y values, as we get large x values and as we get small x values. We can determine the end behavior of any polynomial function from its degree and its leading coefficient. For the large values of x, we can model the behavior of function that behave in the same way.
Similarly, when x approaches to $\infty$, then left part of the graph is also going upward. But, when x approaches to $-\infty$, the left part of the graph extends towards downward direction.
It can also be directly manipulated by having a look at the graph that which part of the graph will go up and which one will go down when the function reaches nearest to the extreme points. End behavior may also be found by applying some rules and formulae on the equation of given function. In the page below, we shall understand about the end behavior of a function and method of its estimation.



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