Common Core Standards: Math
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Students should know that any relationship is all about give and take. For instance, you'll give them an A if they take their homework seriously (or Shmoopily). In any case, you've been giving them functions for far too long now. It's their turn to give some functions back.
Students should know the difference between an arithmetic sequence and a geometric sequence. An arithmetic sequence is a list of numbers in which we add a constant number to the previous one. A geometric sequence is a list of numbers in which we multiply the previous number by a constant called the "common ratio." Basically, arithmetic is addition and geometric is multiplication.
Given a graph of an equation or inequality, pairs of input and output values, and a description of a relationship, students should be able to come up with an algebraic way to represent it. Namely, functions.
It's easiest to start with input and output values. That way, the students can clearly see how x is changing relative to f(x). For instance, given this table of values, what can we decipher? (Do they form arithmetic or geometric sequences?)
The first thing that must be done is for students to realize by how much our independent, or input value, is changing. That is, compare the x values. In this case, it's a relatively easy 1-unit interval.
Next we must look at how much the dependent value (f(x), or the y variable) is changing with respect to its input value. That is, compare the f(x) value to its x value. In this case, once again, it's simply by 1 unit.
But we're not done yet! We must see how our output values change between one another as the independent variable changes. That is, compare the f(x) values to each other. In this case the interval is a steadfast 1. This is an arithmetic sequence.
Students should recall that when the difference in interval is constant, we can presume that our equation is most likely linear. In this case it is simply a matter of f(x) = x + 1.
When graphs are involved, the easiest thing to do is plot points. That way, students can assemble a list of input and output values from the graph. As for descriptions, words to watch out for are "exponential," "linear," "multiple," "constant," and "factor."