Definitions and Formulas

A parameter is a number in the formula for a function that is constant. Changing a parameter will change the entire behavior of the function. The two parameters you are most familiar with are the slope and y-intercept of a linear function. If the slope parameter is changed, the line is more or less tilted; it may even change the direction of the tilt. If the y-intercept is changed, the graph crosses the y-axis at a different point. Most functions come in families of functions that all have the same formula, but the formula has parameters in it. Thus, linear functions of the form should really be called the ”family of linear functions” since there are two parameters in the formula. To get the equation of a specific member of the family, we need to substitute in values for each of the two parameters, A and B. (Just like you need a first and last name to find a specific person in your family; you may sometimes need even more information about the person if more than one person in the family has the same name. Some functions also need more than two parameters. See quadratics below for such an example.)

This is a broad family of functions. The general form of a basic power function is where b is a number. Thus, this family includes the squaring function (b = 2), the square root function (b = 1
2

), the reciprocal function (b = -1), and the basic linear function (b = 1). This family is called the family of power functions because the independent variable, x, is always raised to a power. The shape of a power function depends on whether the power, b, is even or odd. Even power functions look something like a ”U” when graphed. Odd power functions (with b > 1) look more like chairs: on the left they drop off; on the right they rise up high; in the middle they are relatively flat. All basic power functions pass through the origin (0, 0) and the point (1, 1). This is because zero raised to any power is zero and 1 raised to a power is always 1.

A polynomial is a function made from adding together a bunch of power functions that all have whole number powers. (A whole number is a number like 5, 2, 0. Negative numbers and numbers with decimals and fractions are not allowed.) Each power function in a polynomial is multiplied by a coefficient and then they are all added together:

y = anxn + an -1xn-1 + ...+ a2x2 + a1x1 + a0

Notice that since anything raised to the zero power is 1, there is no need to write x⁰ in the last term. Each of the individual combinations of a coefficient and a power function in a polynomial is called a term. Polynomials include several well-known families of functions: the quadratics (see below) and the linear functions:

y = A + Bx.

The n in a power function gives the highest power in the polynomial. It is called the order of the polynomial. The shape of a polynomial function is highly dependent on the order of the polynomial, since this determines the leading power function in the polynomial. The following general statements can be made:

If n is even, then the polynomial function does the same thing on both sides of the y-axis: it either rises up on both sides or drops down on both sides. If n is odd, then the polynomial does the opposite on both sides: one side will rise, the other will drop. The order also determines two other properties: the maximum possible number of times the polynomial crosses the x-axis (the number of zeros) and the number of time the graph changes direction (either from increasing to decreasing or vice versa):

Maximum number of zeros = n
Maximum number of turning points = n - 1

A quadratic function is a second-order polynomial that produces a ”generalized squaring function”. It is usually written in the following way:

2 y = Ax + Bx + C.

You may have seen the famous quadratic formula. This is a formula for finding the roots of a quadratic equation. Roots are places where the function crosses the x-axis, so these points all have y = 0. Thus, they are solutions to the equation:

2 0 = Ax + Bx + C.

Using the quadratic formula, we can find the x-coordinates of these crossing points:

√----------- --B-±---B2----4AC-- x = 2A

Most software can add quadratic trendlines to a graph; however, it refers to them by their more proper name as ”polynomials of order 2”.

Sometimes the data we are trying to fit looks exactly like a basic function, but moved up or down. We can fix this by adding in a vertical shift to the equation. If the graph of a function has been vertically shifted, the graph has the same exact shape, only every single y-value has been increased by the same amount or every y-value has been decreased by the same amount. Effectively, this moves the entire graph of the function either up or down the y-axis. Thus, a vertical shift by k will move the y-intercept up by k units.

Sometimes, the data is moved right or left of the basic function that it is most similar to. We can compensate by adding a horizontal shift to the equation of the graph. If a graph has been horizontally shifted, the graph has been moved to the right or the left. Thus, if the graph is moved to the right h units, then the zeros of the function (if any) will all move to the right h units.

This is the general term to refer to any type of shift (vertical or horizontal).

It is sometimes necessary to stretch a graph out or compress the graph of a basic function so that it will match up better with the data. This can easily be done by multiplying the entire function by a scaling factor.

11.2.1 Definitions and Formulas