# Coefficients

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# 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.

## Linear 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.

## Polynomial Functions

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.

## Power Functions

Power functions are functions in which the variable is the base of a number with a constant exponent. Therefore, when interpreting the graph, the base changes while the exponent stays the same. Therefore, depending on the exponent, the graph could a various collection of curves.

If the exponent is an even number, the graph will be symmetrical about the y-axis. This is because even when x is negative, raising it to an even power will cause it to be positive. Two negatives always cancel each other and make a positive number. If the exponent is odd, the graph will not be symmetrical since when x is negative and raised to an odd power, there will be negative y values on one side of the graph and positive values on the other.

The shape of the graph is also influenced by a few other factors. If the exponent is greater than zero, the graph is increasing and if it is less than zero, it is decreasing.

If the exponent is a fraction $\frac{m}{n}$, both $m$ and $n$ must be examined and their relationship to one another in order to understand what shape they may make. If $n$ is even, then it is undefined for values less than zero. When $n$ is even, there are no negative y values produced. If $n$ is odd than it is defined for values greater than and less than zero (and at zero). If $m$ is even, the function is even (symmetrical) and if $m$ is odd, the function is odd, meaning it is symmetrical about the origin. If $m$ is less than $n$, the function is increasing at a decreasing rate, and if $m$ is greater than $n$, the function is increasing at an increasing rate.

There are other coefficients to consider in a power function. When looking at the general equation, $y = ax^{b}$ we have so far, only discussed $b$. The coefficient $a$ however, is just a scaling factor and therefore makes the graph appear thinner if larger and fatter is smaller. If it is negative, then the graph will be flipped.

## Exponential Functions

Exponential equations are equations where the variable is the exponent, therefore the constant is the base. Although the base isn't necessarily a coefficient, what number it is, changes the shape of the graph. In an exponential function, this base is always a positive number. If the number is between 0 and 1, the graph is decreasing, because one is raising a fraction to higher and higher powers, which makes the number smaller and smaller. If the number is greater than one, the graph is increasing because it is raising the base to higher and higher powers which will make the values increase. If the base is 1, the graph will make a straight line due to the fact that one raised to any power is always one. A coefficient can be multiplied to this base and its variable exponent and will scale the graph. Additionally, one can add a constant to the equation which will shift it upwards or downwards.

## Logarithmic Functions

Logarithmic functions have the general form $y = alog_{b}(cx + d) + e$. The first coefficient, $a$ is just a scaling factor. $b$ is not a coefficient but a base value which affects the shape of the graph. Most of the major differences between the graphs depending on their bases is whether or not the base is less than or greater than one. When greater than one, the graph will be increasing and when less than one, will be decreasing.

## Trigonometric Functions

Trigonometric functions have the general form $y = a f(bt - c) + d$. The $f$ represents any of the trig functions sine, cosine, tangent, secant, cosecant, or cotangent. These functions always include a pair of parentheses in which the independent variable is placed. That is why there is a pair of parentheses in the function and the value $t$ is the independent variable.

$a$, the first coefficient, affects the vertical distance between the function's peaks and valleys. The amplitude is half the distance between the highest and lowest points on a trigonometric graph. Therefore, as it increases, the function increases and if it decreases, the vertical difference decreases, and if it is negative, the graph is flipped over the x-axis.

The second coefficient, $b$ will scale the graph horizontally. It will either shrink or stretch the size of the graph along the x-axis. Larger values will cause it to shrink while smaller values will cause it to stretch. This is because $b$ affects how many times the function repeats its cycle. When the value is negative, it will will flip the graph over the y-axis.

$c$ c will shift the graph right if positive and will shift it to the left when negative. This result occurs because this value is subtracting from $t$.

$d$ will shift the graph up or down. It will shift it up when positive or down when negative, resulting from its addition to the y value.

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