Equations, Functions, and Formulas


What are these expression things good for, anyway? You can't have equations and formulas without them. Without equations and formulas, the whole world pretty much falls apart. Read on, dear Shmooper. Read on.

  • Equations

    Remember when we said equations are like sentences? We have two phrases (expressions) connected by a verb (equal sign). That sure sounds like a sentence to us.

    Equations are made up of two different expressions, one on each side of the equal sign.

    expression number 1 = expression number 2

  • Functions

    Can we be real with you for a sec? A function is probably one of the most important concepts in mathematics. We may say that a lot, but this time we mean it. Functions sort of like machines that take our questions and give us answers.

    A function may take many inputs and outputs of a single number. The inputs are called the independent variables. The number that comes out of the machine is called the dependent variable. The term "dependent" should make sense because whatever comes out of the machine depends on what we put into it. Say you want to make a banana berry smoothie in your blender; you probably shouldn't put raw tuna and Saltines into it. You need to cook the tuna.

    Sample Problem

    Consider the grade that you got in a class. You don't need to tell us what it is—you can keep it to yourself. If it was worse than an A++ , we know you'll try harder next time. Whatever grade you receive at the end of the semester depends on the individual grades you get on each homework assignment, every quiz, and your class participation. Your teacher looks at all of these and determines your final grade.

    The independent variables are the homework scores, the quiz scores, and the extent of classroom participation. The dependent variable is the final grade you get in the class, because it depends entirely on the independent variables. If you got A's on everything you did in class and then received an F for the semester, that wouldn't make any sense. That would be like eating a metric ton of vegetables and then not getting any ice cream. Ripoff. Anyway, the function machine in this example is your teacher's grading policy.

  • Independent and Dependent Variables

    Let's take a closer look at how independent and dependent variables work by working through some examples.

    Sample Problem

    Jim gets paid $10 per hour. The amount Jim gets paid in a week depends on the number of hours he works that week. If all he did was put in two hours shelving books at the library, he'll barely be able to afford to buy a book. Good thing he works at the library.

    On the other hand, if he works 12-hour days in the assembly line of an automobile factory, he can afford to buy all the books he wants. Same rate (input), different pay (output). 

    What's an equation that represents his pay?

    Let's say p represent the amount Jim gets paid in a week, and h represent the number of hours Jim works that week. That means we can set up the following equation:

    p = 10h

    The letters p and h are called variables because they're not fixed numbers. This fact reminds us: be sure to have your numbers spayed or neutered. The quantity h varies because Jim may work a different number of hours each week. The quantity p varies because p depends on h.

  • Formulas

    Here's video to help illustrate a few formulas.

    It's been a loooong time since you've needed to deal with formula on a daily basis. You probably transitioned to solid foods at around five months, but now, it's back. 

    Actually, that formula is gone forever, but mathematical formulas are here to stay forever. Everyone knows that mathematical formulas are twice as delicious, anyway.

    In math a formula, like an equation, is a group of symbols that forms a meaningful mathematical statement. However, we call some equations "formulas" because they help us solve a type of math problem, not just a single math problem.

    The equation A = lw is the formula for the area of a rectangle because it helps us find the area of any and all rectangles, not just one.

    Sample Problem

    Draw a rectangle with side lengths l and w. On second thought, save yourself the trouble—we'll draw one for you. We're really good at rectangles.

    Rectangle with width w and length l.

    Let A represent the area of the rectangle. Then, the equation A = lw is a formula describing A in terms of l and w.

    Sample Problem

    What does the formula for the circumference of a circle C = 2πr tell us to do? 

    The formula says that to find the circumference C, we've gotta multiply the radius r by 2π. The equation is a formula because we can find the circumference of any circle using it.

    Sample Problem

    Stacy came up with the equation 2n + 3y = -1/2 to find the area of a square, and it worked. The formula gave Stacy the correct area of the square, but it didn't work for the next square. Is 2n + 3y = -1/2 a formula for the area of a square?

    Not this time. For an equation to be a formula, it must help find the solution for an entire type of math problem. Stacy created the equation to find the answer of a single problem, so this equation hasn't reached formula status yet.

    Sample Problem

    Is the Pythagorean theorem a formula?

    No...well...kinda...actually, no. No, it's not. But the Pythagorean theorem does lead to a formula. The Pythagorean theorem is a mathematical law about how a right triangle's sides are related to each other, which leads us to the formula a2 + b2 = c2. This formula will help us find the missing length of a right triangle's side if we know the lengths of the other two sides for any and all right triangles.

    There are some formulas that appear frequently in algebra (distance formula, area formulas, etc). It's a good idea to memorize them by heart. Get to know them, ask about their families. You know...the yooj.

    There are also unit conversion formulas which are crazy-useful to know in some problems (Celsius to Fahrenheit, meters to feet, etc.), but it's not absolutely necessary to memorize them.

    Warning: Always know what the variables of a formula represent. Do whatever it takes. Seriously. Write out in the margin of your paper, "H stands for the area of half a circle" or something like that. It's a lot easier to answer questions correctly if you know the meanings of all the symbols you're using. Just as it's easier to avoid getting into a car accident when you know that that red octagonal sign means "stop."