10.11: Linear, Exponential, and Quadratic Models
What if you were given a table of x and y values? How could you determine if those values represented a linear function, an exponetial function, or a quadratic function? After completing this Concept, you'll be able to identify functions using differences and ratios between their values.
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CK12 Foundation: 1011S Linear, Exponential and Quadratic Models
Guidance
In this course we’ve learned about three types of functions, linear, quadratic and exponential.
 Linear functions take the form
y=mx+b .  Quadratic functions take the form
y=ax2+bx+c .  Exponential functions take the form
y=a⋅bx .
In realworld applications, the function that describes some physical situation is not given; it has to be found before the problem can be solved. For example, scientific data such as observations of planetary motion are often collected as a set of measurements given in a table. Part of the scientist’s job is to figure out which function best fits the data. In this section, you’ll learn some methods that are used to identify which function describes the relationship between the variables in a problem.
Identify Functions Using Differences or Ratios
One method for identifying functions is to look at the difference or the ratio of different values of the dependent variable. For example, if the difference between values of the dependent variable is the same each time we change the independent variable by the same amount, then the function is linear.
Example A
Determine if the function represented by the following table of values is linear.




–4 
–1  –1 
0  2 
1  5 
2  8 
If we take the difference between consecutive
Since the difference is always the same, the function is linear.
When we look at the difference of the
For example, examine the values in this table:



0  5 
1  10 
3  20 
4  25 
6  35 
At first glance this function might not look linear, because the difference in the
Another way to think of this is in mathematical notation. We can say that a function is linear if
Differences can also be used to identify quadratic functions. For a quadratic function, when we increase the
Here are some examples of quadratic relationships represented by tables of values:
In this quadratic function,
In this quadratic function,
To identify exponential functions, we use ratios instead of differences. If the ratio between values of the dependent variable is the same each time we change the independent variable by the same amount, then the function is exponential.
Example B
Determine if the function represented by each table of values is exponential.
a)
b)
a) If we take the ratio of consecutive
b) If we take the ratio of consecutive
Write Equations for Functions
Once we identify which type of function fits the given values, we can write an equation for the function by starting with the general form for that type of function.
Example C
Determine what type of function represents the values in the following table.



0  5 
1  1 
2  3 
3  7 
4  11 
Solution
Let’s first check the difference of consecutive values of
If we take the difference between consecutive
To find the equation for the function, we start with the general form of a linear function:
Example D
Determine what type of function represents the values in the following table.



0  0 
1  5 
2  20 
3  45 
4  80 
5  125 
6  180 
Solution
Here, the difference between consecutive




0  0  
1  5 

2  20 

3  45 

4  80 

5  125 

6  180 

When the
To find the equation for the function that represents these values, we start with the general form of a quadratic function:
We need to use the values in the table to find the values of the constants:
The value of
Therefore the equation of the quadratic function is
Watch this video for help with the Examples above.
CK12 Foundation: 1011 Linear, Exponential and Quadratic Models
Vocabulary
 If the differences of the
y− values is always the same, the function is linear.
 If the difference of the differences of the
y− values is always the same, the function is quadratic.
 If the ratio of the
y− values is always the same, the function is exponential.
Guided Practice
Determine what type of function represents the values in the following table.



0  400 
1  500 
2  25 
3  6.25 
4  1.5625 
Solution
The differences between consecutive
Each time the
To find the equation for the function that represents these values, we start with the general form of an exponential function,
Here
Practice
Determine whether the data in the following tables can be represented by a linear function.




10 
3  7 
2  4 
1  1 
0  2 
1  5 




4 
1  3 
0  2 
1  3 
2  6 
3  11 



0  50 
1  75 
2  100 
3  125 
4  150 
5  175 
Determine whether the data in the following tables can be represented by a quadratic function.




10 
5  2.5 
0  0 
5  2.5 
10  10 
15  22.5 



1  4 
2  6 
3  6 
4  4 
5  0 
6  6 




27 
2  8 
1  1 
0  0 
1  1 
2  8 
3  27 
Determine whether the data in the following tables can be represented by an exponential function.



0  200 
1  300 
2  1800 
3  8300 
4  25800 
5  62700 



0  120 
1  180 
2  270 
3  405 
4  607.5 
5  911.25 



0  4000 
1  2400 
2  1440 
3  864 
4  518.4 
5  311.04 
Determine what type of function represents the values in the following tables and find the equation of each function.



0  400 
1  500 
2  625 
3  781.25 
4  976.5625 




3 
7  2 
5  1 
3  0 
1  1 
1  2 




14 
2  4 
1  2 
0  4 
1  2 
2  4 
3  14 
Image Attributions
Description
Learning Objectives
Here you'll learn how to identify a function's type by examining the difference or the ratio of different values of the dependent variable.