Section 3.1
Variables, Constants, and Parameters

Whether a mathematical notation is a variable,  parameter, or constant depends on what you mean by it.

• If you intend to represent the value of a quantity whose measure varies within a situation, then you are using that notation as a variable.
• If you intend to represent the value of a quantity whose measure is the same within all situations (e.g., $\pi$), then you are using that notation as a constant.
• If you intend to represent the value of a quantity that is constant in a particular situation, but which can vary from one situation to another, then you are using that notation as a parameter.

Examine the cylinders depicted in Figure 3.1.1. They are partially filled with water. As they sit, it appears that nothing about the cylinders or water in them varies. Now imagine that each cylinder is being filled from a pipe at its bottom. Then the water’s height in each cylinder varies.

If we let $h_1$, $h_2$, and $h_3$ represent the water’s height in the respective cylinders and imagine that each height varies, then the values of $h_1$, $h_2$, and $h_3$ vary as the water’s height varies within its cylinder. Play the animation in Figure 3.1.1 to see what you envisioned. Move your cursor away from the animation to make the control bar disappear.

Figure 3.1.1. $h_1$, $h_2$, and $h_3$ represent the height of the water in the three cylinders, respectively. Each variable's value varies as the water’s height varies. $r$ and $H$ are parameters. Their values are constant for each tank, but their values vary across tanks.

Some quantities in Figure 3.1.1 do not vary within a cylinder but vary across cylinders. Any cylinder’s radius does not vary, nor does its height, nor does the area of its water’s exposed surface. For each cylinder, its water’s height and volume do vary. So if we let r represent a cylinder’s radius, the value of r, for each cylinder, does not vary. But the value of r varies across cylinders. Similarly, if we let H represent a cylinder’s height, the value of H, in each cylinder, does not vary. But the value of H varies across cylinders.

Again, when we say that a notation has the meaning of a variable, we mean that it represents the value of a quantity whose value varies within a situation. When we use a notation to represent a value that is constant in a situation but which can vary from one situation to another, we are using the notation as a parameter. We mean that the notation represents the value of a quantity that is constant within a situation, but the quantity could have different values in different situations.

By convention, we will use the term domain to refer to the set of values that a variable can have. If in Figure 3.1.1 we use b to represent the initial height of water in a cylinder, then the domain of $h_i$ for any of i = 1, 2, or 3, is b ≤ $h_i$ ≤ H.The letters b, r, and H are used as parameters. The letters $h_1$, $h_2$, and $h_3$ are used as variables.

In this textbook, we will assume that the domain of any variable is the set of real numbers unless stated otherwise or unless restricted by context.

We also can speak of a variable’s value varying in the absence of a specific quantity. When we say something like, "The value of x varies between 2 and 5", we are actually saying that there is some yet-to-be-known quantity lurking in the background whose measure varies from 2 to 5, passing through all real numbers between 2 and 5.

Values that do not vary are called constants. Some constants are universal, like π and e. Their values do not depend on a particular context. Other constants are constant only with respect to a situation. In the statement, "Billy has 5 toys. His father gave him 2 more. He has 5+2 toys altogether", the number of toys Bill started with and the number of toys he received are constants.

In the statement, "Billy has m toys. His father gave him n more. He has m+n toys altogether", the number of toys he started with and received are constants within the situation. But they are expressed generally. This means that the statement is given with the intention that we are describing many potential situations. The statement " m+n" says what to do with those numbers once they are known. The letters "m" and "n" are used as parameters.

Figure 3.1.2 summarizes the distinctions among variables, parameters, and constants.

Figure 3.1.2. Types of notation and their meanings. What a notation means depends on what you intend it to represent.

The word "Change" is ambiguous, so we use more precise language

Ways You May Envision a Variable's Value Varies

There are, in principle, three ways you can envision a variable's value varying:

• Discretely (one value, then another, then another) such as the variable $x_1$ in Figure 3.1.3.
• Chunkily, like laying down rulers end to end. The magnitude of the variable's values changes from one value to another, but we also get all values within a chunk without passing through them (see variable $x_2$ in Figure 3.1.3).
• Smoothly. The magnitude of the variable's value varies continuously through its domain. No chunks, no gaps, just smooth variation through values, as in variable $x_3$ in Figure 3.1.3

Unless stated otherwise, for any variable whose value varies, presume its value varies smoothly.

Exercise Set 3.1

1. You have 240 meters of fence to enclose a rectangular lawn. You are free to make the enclosure have any possible length and width, but you must use all the fence. Play the animation. Move your cursor away from the animation to make the control bar disappear.

   
   
   1. Define constants in this situation. Use parameters for constants whose values you do not know.
   2. Define variables in this situation. State the intervals over which they vary.
   3. Express the relationship between the rectangle’s width and length symbolically.
   4. Express the relationship between the rectangle’s enclosed area and either its width or its length.
2. A droplet of water lands in a large bowl of water, creating circular ripples. The ripples expand to a radius of 2 meters before dissipating. Play the animation. Move your cursor away from the animation to make the control bar disappear.
1. Define constants in this situation. Use parameters for constants whose values you do not know.
2. Define variables in this situation. State intervals over which they vary and the nature of their variation.
3. Express the relationship between the outermost ripple’s radius and its enclosed area symbolically. What constants and variables are in this relationship?
4. Express the relationship between the the outermost ripple’s enclosed area and the number of seconds since the droplet hit the water symbolically. What constants and variables are in this relationship?
3. When we represent a linear function by the statement y = mx + b, which letters are variables? Which letters are parameters? What makes some letters parameters and other letters variables? What is the nature of their variation?
4. The figure below shows an animated graph of y = mx + 1 for m between -4 and 4. Explain how this animation works in terms of the meanings of constants, variables, and parameters. You will need to talk about how a graph is made. Move your cursor away from the animation to make the control bar disappear.


5. State three ways in which someone could interpret the statement, "The value of x changes from 2 to 7."
6. Re-write each of these statements three times using forms of "vary", "variation", or "substitute", once to reflect the meaning "change in progress", once to mean "completed change", and once to mean "have a new value".
   1. The value of y changed from 3 to 9 as the value of x changed from 1 to 4.
   2. x was 3 and later it was 7
   3. The cylinder's volume changed by 3 ft³ as its height changed by 0.5 ft.
7. Consdier this list of quantities whose values vary. Decide the way you envision their values varying in preparation to discuss it
   Let N represent the number of students inside your classroom prior to the beginning of class. How do you envision the value of N varying?
   Let M represent the measure of a room's width as you lay meter sticks one end to the other. How do you envision the value of M varying?
   Let P represent the measure of a room's width as you pull a measuring tape with the free end attached to one wall. How do you envision the value of P varying?
   Let V1 represent the volume of water in a tank as you fill it by repeatedly pouring water from a bucket. How do you envision the value of V1 varying?
   Let V2 represent the volume of water in a tank as you fill it from a hose connected to a city water outlet. How do you envision the value of V2 varying?
   Describe a novel situation in which the primary variable varies discretely.

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