7.3: Identifying Functions from Numerical Patterns

So, before I begin, I'd just like to say that Blogger is very Mac-intolerant. Because I myself am a Mac user, Blogger.com gives me less formatting options (I only have spellcheck and insert a picture). There are several tables throughout this blog, but the only way i could get them to separate is by a series of dots. Please forgive this slight amount of sloppiness.


This lesson is all about identifying the type of function when given a set of data. Each function has a specific pattern between the x and y values which will help you determine the equation.

Chapter Summary

Linear
Linear equations are probably the easiest to identify (but i'm sure you knew that already). If you look at a set of data, such as,

x .......y
1 .......7
2 ...... 9
3 ......11
4 ......13
5 ......15

It can be determined that the following set of data is a linear function due to the fact that both the x and the y values increase in a consistant, addition manner. When each x value increases by one, the y value increases by two. Because both sides increase by addition, this is said to be an ADD-ADD PATTERN.

Power
Both the x and y values in a power function increase raipdly, due to the fact that both values are increased in muliplicational incriments. Here is a set of points in a power function.

x.......... y
3........135
6....... 1080
9 .......3645
12 ......8640

Looking at the x values first, the first increase goes up by an addition rate of +3, but the y values do not share consistency with this pattern. therefore, we must move on to multiplying both values by a certain number. in this situation, when the x value is multiplied by two, the y value is multiplied by a value of eight (or, rather, two cubed.) Because you must multiply both sides, power functions are said to be MULTIPLY-MULTIPLY PATTERNS.

Exponential
Exponential functions are, in a way, a blend of the Linear and Power functions in the sense that, though the x values increase from addition, the y values increase through multiplication.

x .............y
1 ............15
3 ............135
5 ..........1215
7 ........109355

We see here that, while the x values go up by an addition increment of +2, the y values go up by a multiplication value of 9. Because of the addition, there is still some consistancy between the increasing y values. Due to the mix of addition and multiplication, exponential functions are referred to as ADD-MULTIPLY PATTERNS.

Quadratic
Quadratic functions are a bit more complicated; the similar intervals required are not between each y value, but the difference of each interval of the y values (I hope this isn't confusing you more... it's hard to explain, but I'm doing my best... bear with me, people.) Take this set of x and y values, for example.

x............... y
1 ..............15
3 ...............5
5 ..............19
7 ..............57
9 .............119

The first thing that we can quickly notice is that the x values increase in a simple addition matter, each variable being two greater than the previous one. As for the y values, the difference between the first two values listed is a decrease of ten (or an increase of -10). The second and third values listed are fourteen numbers apart from each other. These two values of -10 and 14 are at a value of twenty four apart from each other. the third and fourth y values are 38 apart from each other, while the fourth and fifth values are 64 away from each other. These two values share a difference of 24. The second difference of the values is always constant, so Quadratic functions are said to be CONSTANT SECOND DIFFERENCE PATTERNS. If this information is not enough to get rid of any confusion you have, feel free to ask Mr. French or myself at any time.

NOTE: One clue as to determining whether or not a set of values belong to a quadratic funtion is searhing for a change in the direction of the y values. For instance, in the example given, the first two values are 15 and 5. This data tells us that the funtion is decreasing. The third value, 19, moves is a positive direction from its previous value. Quadratic functions are the only functions in which you will see a change in y value direction.




You may also be given to f(x)s, and then asked to list the possible outcomes. Use your knowlege of what each function can be described as (add-add, multiply-add, etc.) to determine how the variables are changed in order to receive the following values.

For example...

f(5) = 12
f(10) = 18
f(20) = ?


You must allow yourself to look at these given equations and solve them from various viewpoints. You can say that x increases +5 between the two equations. The change can also be interpreted as x2. Similarily, the y values can be seen as changes of +6 or x1.5. By working out these different possibilities, f(20) can equal several things...

...in a linear equation, f(20) = 30

...in a power function, f(20) = 27

...in an exponential function, f(20) = 40.5



Also, verbal explanations are key in what an equation looks like.

If y varies directly with x,
-if x doubles, y doubles

If y varies inversely with x,
-if x doubles, y is reduced by .5

If y varies directly with the cube of x,
-if x doubles, y is multiplied by eight (which is equivalent to 2 cubed)


Example Problem

Question
Using the given points, determine wether the data is an add-add, add-multiply, multiply-multiply, or constant second differences pattern. Tell what kind of function that has this pattern. What should the next three sets of values be?


x..........f(x)
2...............14
6..............112
18............896
54............7168


Solution

1) First of all, we need to look at the list of x values. There is not any addition consistancy between them. Once multiplying is tried, we find that each value is multiplied by a factor of 3 to get the next value.

2) As for the y values, there too is no addition link. We divide the second value, 112, by the first value, 14, to get 8 (or two cubed). This is also the answer you get when you divide 896 by 112, and 7168 by 896.

3) So we know know that this is a Multiply-Multiply pattern, which tells us that we are dealing with a power function.

4) Since we figured out that when each x value is multiplied by three, the y value is multiplied by 2 cubed, we can determine the next three vaules in this pattern. The first x value is found by multiplying the previous x value, 54, by 3, to get 162. The x value after that is found with the same means, and 162 times 3 is 486, and repeating the process once more gives us the next x value of 1458.

5) As for the y values, we must multiply the last given y value, 7168, by 2 cubed, or 8, in which case we get 57344. Repeating this process twice more gives us the values 458752 and 3670016.

6)
So the final table looks like this...

x..........f(x)
2...............14
6..............112
18............896
54............7168
162.........57344
486........458752
1458.....3670016


If you need help, here is an extremely extensive website with information regarding these concepts.
Concepts Help



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