Coefficients

(Difference between revisions)
 Revision as of 15:41, 8 November 2012 (edit)← Previous diff Revision as of 10:07, 12 November 2012 (edit) (undo)Next diff → Line 10: Line 10: For other polynomials (functions that use addition, subtraction, and multiplication, and contain terms with integer exponents that are not negative) of higher degrees (the value of the highest power), the graph will be a curve. If the degree is an even number, the graph will be symmetrical, if not it will will have its ends going in opposite directions. The leading coefficient (coefficient of the term with the highest power) will affect this curve. The larger it is, the thinner the curve will appear because it is scaling the entire graph, increasing the y values more and more in comparison to the x value that was initially used. As the coefficient decreases, the curve becomes wider. Whether it is positive or negative also affects the graph. Even degrees, for a positive coefficient, will go up towards positive infinity on either side and if negative, both ends will go towards negative infinity. If the degree is odd, and the leading coefficient positive, the right side will go to positive infinity and the left towards negative infinity, when the leading coefficient is negative, the end behavior of each side will reverse. For other polynomials (functions that use addition, subtraction, and multiplication, and contain terms with integer exponents that are not negative) of higher degrees (the value of the highest power), the graph will be a curve. If the degree is an even number, the graph will be symmetrical, if not it will will have its ends going in opposite directions. The leading coefficient (coefficient of the term with the highest power) will affect this curve. The larger it is, the thinner the curve will appear because it is scaling the entire graph, increasing the y values more and more in comparison to the x value that was initially used. As the coefficient decreases, the curve becomes wider. Whether it is positive or negative also affects the graph. Even degrees, for a positive coefficient, will go up towards positive infinity on either side and if negative, both ends will go towards negative infinity. If the degree is odd, and the leading coefficient positive, the right side will go to positive infinity and the left towards negative infinity, when the leading coefficient is negative, the end behavior of each side will reverse. + + Although the shape of polynomials stays essentially the same regardless of the other coefficients, the remaining terms do affect the shape close to the vertex. This is because the first term becomes significantly larger as the values of x become large positive or negative numbers. Since this value grows larger and larger, the other terms affect the graphs less and less until they are often negligible. In most cases, if one "zooms out" on any polynomial, the little bumps and curves that may be created by other terms in the middle are difficult to notice, because they are so small in comparison to the overall graph, which is infinite in size. To see how the other terms do affect behavior closer to the vertex, see a more mathematical explanation. |Field=Algebra |Field=Algebra |InProgress=Yes |InProgress=Yes }} }}

Basic Description

Every graphical expression has variables, numbers that change dependent upon the location of the point on the graph, and has coefficients and constants, numbers that do not change, but still influence the shape of the graph. The following paragraphs will be examining how these coefficients affect various types of graphical functions.

A single line is a representation of a linear expression shown by the general equation $y = ax + b$. $x$ and $y$ are the variables (the changing values described above); And $a$ and $b$ are the coefficients. $a$ is a coefficient with an $x$ in its term, this means that when $a$ is changed, it will affect both $x$ and $y$. In this case, $a$ affects how quickly one changes with respect to the other. When a is one, these two variables change the same amount in comparison to other, therefore the steepness of the line will be a one-to-one ratio. When $x$ is 5, $y$ is 5. When $x$ is -1.672, $y$ is also -1.672. If $a$ is greater than 1, than $y$ is increasing more rapidly than $x$ and the line becomes steeper, ($y$'s value will be greater than its respective $x$ value). If it is less than 1, but greater than zero, $x$ has greater respective values, therefore the line becomes less steep. If $a$ is zero, $y$ is never changing, it remains at a constant value throughout. Therefore, the line is horizontal. To make the line vertical, $a$ would have to be very very large, at infinity. When $a$ is negative, the line will tilt in the other direction. When $a$ is positive the line goes from the bottom left to the upper right, so when it is negative it goes from the bottom right to the upper left. And as it becomes more negative, it will increase the steepness in this direction.

The coefficient $b$ is specifically a constant (a term without variables). Constants affect where the linear function intersects the y-axis (typically the vertical axis). For instance if $b$ is 1, the line will pass through the part of the y-axis where $y$'s value would correlate to 1. As can be inferred, $b$ only changes the y values. It raises or lowers the line by the value it is equal to.

For other polynomials (functions that use addition, subtraction, and multiplication, and contain terms with integer exponents that are not negative) of higher degrees (the value of the highest power), the graph will be a curve. If the degree is an even number, the graph will be symmetrical, if not it will will have its ends going in opposite directions. The leading coefficient (coefficient of the term with the highest power) will affect this curve. The larger it is, the thinner the curve will appear because it is scaling the entire graph, increasing the y values more and more in comparison to the x value that was initially used. As the coefficient decreases, the curve becomes wider. Whether it is positive or negative also affects the graph. Even degrees, for a positive coefficient, will go up towards positive infinity on either side and if negative, both ends will go towards negative infinity. If the degree is odd, and the leading coefficient positive, the right side will go to positive infinity and the left towards negative infinity, when the leading coefficient is negative, the end behavior of each side will reverse.

Although the shape of polynomials stays essentially the same regardless of the other coefficients, the remaining terms do affect the shape close to the vertex. This is because the first term becomes significantly larger as the values of x become large positive or negative numbers. Since this value grows larger and larger, the other terms affect the graphs less and less until they are often negligible. In most cases, if one "zooms out" on any polynomial, the little bumps and curves that may be created by other terms in the middle are difficult to notice, because they are so small in comparison to the overall graph, which is infinite in size. To see how the other terms do affect behavior closer to the vertex, see a more mathematical explanation.

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