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admin | Category: Electile Dysfunction 2016 | 01.04.2015
Graphs of PolynomialsWhen graphing a polynomial, the number of different real roots is the number of times the polynomial crosses the x-axis. ExamplesHere is an example of a third degree polynomial with three real roots and a positive leading coefficient.
Even DegreeEven-degree polynomials either open up (if the leading coefficient is positive) or down (if the leading coefficient is negative).
Example 1Which of the following could be the graph of a polynomial whose leading term is "-3x4"?
SummaryThe end behavior of polynomials is as follows: For even degree polynomials, with a positive leading coefficient, as x approaches infinity, f(x) or y approaches infinity and as x approaches negative infinity, f(x)or y approaches infinity For odd degree polynomials, with a positive leading coefficient, as x approaches infinity, f(x) approaches infinity and as x approaches negative infinity, f(x) approaches negative infinity Polynomials with negative leading coefficients will have the opposite end behaviors. Example 1An even polynomial of degree higher than two will open up or down, but may contain more than one curve. Example 3An odd polynomial of degree higher than three will look much like a cubic, and may contain more than one curve. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. To predict the end-behavior of a polynomial function, first check whether the function is odd-degree or even-degree function and whether the leading coefficient is positive or negative.
A sketch of the graph can be determined by looking at the equation when you have an understanding of these important characteristics. If the degree is odd, the polynomial will have at least one real root and up to as many as the degree of the polynomial.
If they start lower left and go to upper right, they're positive polynomials; if they start upper left and go down to lower right, they're negative polynomials.


Depending on whether they open upward or downward, they will have a maximum or minimum point.
Since odd functions have opposite end behavior, there is never an absolute maximum or minimum.
Factoring polynomials will be discussed in a later section, however, recall from the quadratic unit, the factored form of a function is y = (x - R1)(x - R2). So, the sign of the leading coefficient is sufficient to predict the end behavior of the function. For example, if the degree of the polynomial is five, there can be as few as one or as many as five real roots or x-intercepts. To write polynomials in factored form, there will be the same number of factors as there are roots. Other names for roots are solutions, x-intercepts, and zeros.Starting with a polynomial of degree 1, this is a straight line that will cross the x-axis 1 time. The rule states that a cubic can have at most three real roots, but it can have less than three.
The exponent says that this is a 4th degree polynomial, so the graph will behave roughly like a quadratic (up on both ends or down on both ends). The exponent says that this is a 7th degree polynomial, so the graph will behave roughly like a cubic: the ends will point in opposite directions.
The largest possible number of minimum or maximum points is one less than the degree of the polynomial. Therefore a 4th degree polynomial might be written as P(x) = (x - R1)(x - R2)(x - R3)(x - R4). The sign of the coefficient of x tells us if the line will be rising or falling as it runs from left to right.


Remember that once a polynomial is factored, setting each factor equal to 0 and solving will determine the roots or x-intercepts. When graphing, sometimes the zeros can be determined just by looking at the graph or table for the function. Since the sign on the leading coefficient is negative, the graph will be down on both ends.
Since the sign on the leading coefficient is positive, the graph will be increasing from lower left to upper right.
Multiply the polynomial to get -x3 + 2x2 - 5x + 10 or (-x)(x2) to find that the leading coefficient is -1.
The leading coefficient is positive 1, which implies the graph will go from bottom left to upper right. Again, the leading coefficient, which in this case is the x3 term, determines which way the graph will point. If it is positive, then the graph will go from the lower left corner to the upper right corner of the grid, while a negative coefficient will give a graph going from the upper left corner to the lower right corner of the grid. If the polynomial has degree n then there will be at most n - 1 turning points in the graph.



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