Characteristics of Linear, Quadratic, and Exponential Functions:

Some Examples

 

 

            This handout explores linear, quadratic, and exponential functions through a few examples.  The treatment is not designed to be complete, but rather to give an accessible introduction to some functions that commonly arise in problem-solving activities.  Such functions can be represented by tables, graphs, recursive and explicit equations, as well as by verbal and pictorial descriptions -- flexibility in moving between such representations is a key algebraic thinking skill.

 

Linear Functions:

 

Example: A Stadium Problem

 

            The first six rows of a stadium section are shown below.  The rows continue the same pattern, and there are 50 rows total.

 

How many seats are there in Row R? 

 

We shall let S represent the number of seats in row R.  We can make the following table of R and S:

 

R = row number           S = Number of Seats in Row R

 

                        1          1

                        2          3

                        3          5

                        4          7

                        5          9

                        6          11

                        7          13

As the row number increases by 1, the number of seats increases by 2.  This pattern is implicit in the way the problem was designed  each row of seats contains two more than the one before it.

 

One way to characterize linear functions is that they can be written in table form so that the differences in both the input and output columns are constant.  Sometimes in problem solving, it’s helpful to mark the differences explicitly:

 

R = row number           S = Number of Seats in Row R

 

                        1          1

     2-1=1                               3-1=2

                        2          3

     3-2=1                               5-3=2

                        3          5

            1                                 2

                        4          7

            1                                  2

                        5          9

            1                                  2

                        6          11

            1                                  2

                        7          13

 

Note that in most situations, a table only includes some of the values of a function.  The characterization says that a linear function can be written as a table in this form, but if we include different values of the function, the table might seem to have different properties.  Here are some other examples of tables of values taken from this same function:

 

R          S                                  R                      S                                  R          S

           

            1          1                                  1                      1                                  5          9

2                                  4          5                                              10            -2                        -4

            3          5                                  6                      11                                3          5

2                                  4          5                                              10            17                       34

            5          9                                  11                    21                                20        39

2                                  4          5                                              10            10                       20

            7          13                                16                    31                                30        59

2                                  4          5                                              10             -19                    -38

            9          17                                21                    41                                11        21                   

Notice that in the first two tables, the input and output differences clearly remain constant, but they are different constants than in the original example.  The last table is unorganized, and the pattern is not clearly visible, although the values are all accurate for rows of the stadium problem.        

            You might have noticed another pattern that holds for all three tables: the increase (or decrease) on the right side of the table is always double the increase (or decrease) on the left side of the table.  In the rightmost table, for example, the input (R.) column first decreases by 2, and the output (S) column decreases by 4, and 4 is twice 2.  Then the R column increases by 17, and the S column increases by 34, or twice 17.

            The ratio of the increase in the output column to the corresponding increase in the input column is called the slope of the function.  Traditionally, the slope concept was introduced in graphical and explicit equation representations of linear functions, but for many students, this concept is most intuitive when presented in the table representation of a function.

 

Exercises (answers are in the back):

 

1.  Determine whether each of the following tables represents a linear function (if the values in the table are consistent with a linear function, you may assume that the whole function is linear  i.e. that the function doesn’t behave differently on values not represented in the table).    

 

      x           y                            a     b                           in          out               p          q

a.                                 b.                                 c.                                 d.

      0            4                           0      17                          0            2                   7          22

        1          9                           3      11                          1            4                   2          7

        2          14                         6      5                           2            8                   3          10

        3          19                         9      -1                           3            16                 0          1

        4          24                         12    -7                           4            32                 5          16

        5          29                         15    -13                         5            64                 6          19

 

2.  Find the slope of each of the linear functions in part 1.

 

 

Recursive Equation Representation:

 

            A recursive equation (also called a recurrence relation) is an equation whose next value depends on previous values.  A recursive equation is a natural representation of the stadium problem.  We can describe the number of seats in row R as follows:

 

            The number of seats in Row 1 = 1

            The number of seats in Row R = The number of seats in Row (R-1) + 2

           

Giving the number of seats in Row 1 is an example of an initial condition.  We need to know how to start our table.  Once we have an initial condition, we can create the next row from the previous row.

            Recursive equations and tables can be highly inefficient methods for solving problems  in order to find the number of seats in row 50, using either representation, we must make a table or compute values from rows 1 to 49 first.  With technology, however, recursive equations can become efficient calculating tools; they are easily entered in both spreadsheets and most graphing calculators, which can then be used to quickly compute large values.  Thus, many topics can become accessible to students who have not yet mastered algebraic notation or graph interpretation (and recursive equations and tables can be used to make connections to help students understand other representations).

            In Excel, we can enter the above recursive equation by first typing a 1 in cell A1, to represent the initial condition.  Then, in cell A2, we can type “=A1+2,” which will add 2 to cell A1 to compute the value in cell A2.  Then, if we drag the formula down from cell A2, it will automatically update, and we can easily drag it down 50 cells to compute the number of seats in any row of the stadium. 

            More formal mathematical notation for the recursive equation looks like this:

                                                                       

or

 

This notation can seem intimidating, but it is almost identical to description with words given above.

            When represented as an explicit equation, a linear function takes the following form:

                        Initial Row = ____   (fill in blank with a number)

 

            Next Row = Previous Row  +  ___    (fill in blank with a number)

 

More formally,

 

,

 

where  and  can represent any numbers (including negative numbers, so the second condition can look like subtraction instead of addition; also we can have  ).  Note that the above representation starts with , so that the initial condition above is given to start at input = 0, but we can actually start anywhere.

 

Exercise:

 

3. Represent each of the functions in problem 1 as a recursive equation (even if the function isn’t linear).  Explain how to tell from the recursive equations whether the function is linear.

 

Graph Representation

 

            A linear equation is called linear because its graph is a straight line.  In the case of the stadium problem, the graph is actually a series of points that are on a straight line  the function that determines the number of seats in each row is not defined if the rows are not natural numbers (i.e. it’s defined if the row number is 1, 2, 3, 4, etc. but not if the row number is 1/2 or 6). 

 

 

Note that according to the everyday definition of slope (i.e. the amount a line is slanted), this graph can appear to have different slopes, depending on how the axes are scaled:

 

                    

 

Both of the above graphs represent the data in the staircase problem.  There is no one “right” way to scale the axes; what’s most important is to recognize how scaling can make graphs misleading.

            As it is defined mathematically, the slope represents the vertical increase when the horizontal value increases by 1.  We can see on the graph, as we did in the table, that when the row number (horizontal value) increases by 1, the number of seats (vertical value) increases by 2.  This relationship remains true no matter how the graph is scaled (although it can be harder to detect accurately depending on the graph).

 

 

           

Explicit Equation Representation:

 

            Most traditional high school algebra courses focus primarily on manipulating explicit equations (which they often simply refer to as “equations”).  An explicit equation is an enormously efficient way to code a lot of information. 

            To find our explicit equation, we will start by extending the graph to a straight line, i.e. by extending the domain (set of possible input values) to include all real numbers (i.e. all numbers on the number line).  Not all points on this line will represent rows of the stadium, but our equation will represent all points on the line (i.e. we will be able to, for example, substitute 1/2 for a row number, and although the answer won’t tell us anything about the stadium problem, it will give us a point on the line). 

            Note that the graph (see next page) starts at rownum=0, and for rownum=0, we have numseats= -1.  It doesn’t make much sense to say that row 0 has 1 seat in it, but if we keep extending the table or graph according to the same pattern, it makes sense that

1 would correspond to 0 in this way.  The value 1 is known as the y-intercept; it is the place where the graph crosses the vertical axis.

            Here’s how looking at the y-intercept can help us.  Let’s start with an example.  Suppose we want to find the number of seats in row 15.  We will start with the mythical row 0, where there is 1 seat.  Now, to get to row 15 from row 0, we move 15 rows (in the stadium itself, we are moving back; on the graph, we are moving right), and we know that for every row we move, we will add two seats, so that we will add 30 seats to our original 1 seat, for a total of 29 seats in row 15.  You can use any of the other representations to check that this answer is correct.

            If the 1 makes you uncomfortable (and it very well might), we can instead start at row 1, with 1 seat, and move 14 rows.  In moving 14 rows, we will add 28 seats, for a total of 29 seats.

            Now, to find the number of seats in row r, we can start with row 0 and move r rows.  Following the logic of the example, we see that

 

            number of seats in row r =  -1  +  2 r   (remember that 2 r means 2 times r)      

      

 

Following the logic of the second example, we see that

 

number of seats in row r  =  1  +  2 (r-1), and by the distributive property, we have

number of seats in row r =  1  +  2r  -2  =  -1 +2r  (just as before)

 

In algebra, you might remember describing linear equations in the form .  In this notation, b represents the y intercept (-1 in our case), m represents the slope (2 in our case), x represents the input or horizontal values (the row number in our case), and y represents the output or vertical value (the number of seats in row r).

 

Exercises:

 

4.  For each of the following linear functions, provide the missing representations (table, graph, explicit, and or recursive equation).

 

a.   

 

b.   

 

c.  A taxi cab fare costs $1.50 when you step into the cab and $.25 for each 1/8 of a mile that you go.

 

d.                                 X         Y

 

                                    -9         20

                                    -6         17

                                    -3         14

                                    0          11

                                    3            8

                                    6            5

                                    9            2

 

e.

             

 

 

Quadratic Functions:

 

Quadratic functions are more difficult than linear functions, and we will not describe them in as much detail in this handout.  Even a little knowledge about quadratic functions can be helpful in problem solving, however.  We will look at three more problems related to the stadium seating; these problems can be described by quadratic, rather than linear, functions.

 

·        What is the seat number of the last seat in row r?

·        What is the seat number of the middle seat in row r?

·        What is the seat number of the first seat in row r?

 

We begin with the first question.  A reasonable start is to make a table:

 

 

 

 

 

 

 

 

R = row number           L = Number of last seat

 

                        1          1

                        2          4

                        3          9

                        4          16

                        5          25

                        6          36

                        7          49

                        8          64

 

As before, we can make a difference table:

 

R = row number           L = Number of last seat

 

                        1          1

     2-1=1                               4-1=3

                        2          4

     3-2=1                               9-4=5

                        3          9

            1                                 7

                        4          16

            1                                  9

                        5          25

            1                                  11

                        6          36

            1                                  13

                        7          49

 

Note that in this case, the differences on the right form a linear function (as opposed to when we looked at linear functions, and the differences were all constant).  When the difference table is equally spaced on the input side, with differences that form a linear function on the output side, the function is quadratic.

            We won’t go into quadratic functions in depth, but one important thing to know about them is that when represented as explicit equations, they include a term that is squared (i.e. multiplied by itself).  In fact, looking more carefully, we can easily see that L = R x R = R², e.g. the last seat in row 6 is 6 x 6 = 36.  The picture below demonstrates why the differences in this table form a linear function:

 

 

If we compare the 3 x 3 square and the 4 x 4 square visually, we see that the difference forms the rotated L shape, and this L shape includes two more square tiles than the previous difference, which corresponds to the differences in the table increasing by 2 each time.. 

            Notice that the differences form the same linear function that we defined in the first stadium problem, i.e. 2r-1.  We can use this information to write L, the number of the last seat in row r, of the stadium problem as a recursive function:

 

 

When represented as an explicit equation, a quadratic function takes the following form:

                        Initial Row = ____   (fill in blank with a number)

 

            Next Row = Previous Row  +  ___    (fill in blank with a linear function)

 

More formally,

 

.

 

The  represents the initial condition, and  represents the general form of a linear function.  The variables  and  can be any real numbers (including negative numbers and 0), but the variable  cannot equal 0, otherwise the recursive equation would represent a linear function, not a quadratic function.

We can use the recursive equation for the last seat in the row to set up a table in Excel as follows (formulas used in row 4 of the table are listed below the variables; the numbers in row 3 of the table are initial conditions and must be entered explicitly):