CHAPTER 14 Algebraic Thinking: Generalizations, Patterns, and Functions

FUNCTIONS  A function is a relationship between two sets of data.  These two sets are called the domain and range.  Elements of the domain are called inputs and elements of the range are called outputs.  The input variable is called the independent variable whereas the output variable is called the dependent variable.  A function must have a unique output for each input in the domain.  A function that does not repeat outputs among different inputs is called one to one.

EXAMPLES  Give two examples of a function and two examples of a non-function. For your first pair of examples, you may use numbers. However, give non-numerical examples for the second pair.  Give two examples of a one to one function and two examples of a function that is not one to one. For your first pair of examples, you may use numbers. However, give non-numerical examples for the second pair.

DIFFERENT REPRESENTATIONS OF FUNCTIONS  Consider the following example: Brian sells hotdogs during basketball games. He pays the cart owner $35 per night to use the cart. He sells hots dogs for $1.25 each. His costs for things like the hot dogs, buns, condiments, napkins, etc. are about $0.60 cents per hotdog on average.  Context: A carefully chosen context can foster understanding of functions. What is our context here?  Table: We can make a table for the various hot dogs sold and the corresponding profit. Make a table with a few values for the independent and dependent variable.  Verbal Description: Here we have profit as a function of hot dogs sold. How would we describe the function in words using the numbers given?  Symbols: Symbols allow us to define an equation that provides the mathematical relationship between the variables. What is the equation here for profit p?  Graphs: If we can plot a graph, we are able to visualize how the function changes over the independent variable. Make a graph now with hot dogs sold on the horizontal axis and profit on the vertical axis. Make sure everything is labeled.

DIFFERENT REPRESENTATIONS OF FUNCTIONS  Consider the following example: Brian sells hotdogs during basketball games. He pays the cart owner $35 per night to use the cart. He sells hots dogs for $1.25 each. His costs for things like the hot dogs, buns, condiments, napkins, etc. are about $0.60 cents per hotdog on average.  What is the breakeven point? In other words, how many hot dogs must be sold to profit $0?  What is the number of hotdogs that must be sold to profit $100?

LINEAR FUNCTIONS  Linear functions have constant growth. There importance cannot be understated so it’s crucial that they are explored in a variety of ways.  To be linear, any incremental change in the input variable, must result in a constant change in the output variable.  As a result, we have that the (change in output)/(change in input) = constant.  Often we write, y = mx + b where m represents this constant slope and b represent the y-intercept.

LINEAR FUNCTION PROBLEMS  Consider the two points (2,3) and (-3,4).  Plot these points and the line that passes through them. What is the slope of this line?  Write the equation of the line in the form y = mx+b.  What is the equation of the line that is parallel to the above line, but also passes through the point (5,6)?  What is the equation of the line that is perpendicular to the first line above that passes through the point (5,6)?  In general, how are the slopes of perpendicular lines related and why?

QUADRATIC FUNCTIONS  Quadratic functions are characterized as functions that have constant second differences. In contrast, linear functions have constant first differences.  Example: Suppose that f(x) = x^2+x+1.  Compute the first differences? Are they growing at a constant rate?  Compute the second differences? Are they growing at a constant rate?  They take the form f(x) = ax^2+bx+c.  They can also be written in the form f(x) = a(x-h)^2+k where (h,k) is the vertex of the corresponding parabolic graph.  In context, linear functions can represent linear or one-dimensional measures whereas quadratic functions can represent area or two- dimensional measures.

QUADRATIC FUNCTION PROBLEMS  Factor the function x^2+5x+6 into two linear factors and draw the two-dimensional model that represents the product of these factors. Pretty cool, huh?  Suppose that price of bananas is 22 cents per banana when you sell 200.  Suppose that for every 1 cent increase, you sell 5 less bananas. What is the linear relationship between price and quantity?  Now, use this linear relationship to build a revenue model. Indeed, revenue is p x q where p is price and q is quantity. Use the previous problem to express revenue in terms of price only. This should be a quadratic model.  Finally, determine the best price to sell these bananas and the total amount of money collected. To do this, we might consider completing the square.  Suppose you are building a rectangular pen with width W and length L for your cows that will have a fixed perimeter of 100 feet. Explore the different representations of the function that expresses area as a function of width, i.e., build a table and then plot the graph.  Consider the sequence 2, 6, 12, 20, … Compute the first and second differences. Write the function f(n) that yields these values for n=1, 2, 3, …