function rules
- Thread starter cgates67
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mmm4444bot
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I'm confident that Denis intended to type "sequence" instead of "series".
You could also try "arithmetic sequence progression" as keywords to get even more relevant hits.
Your function will be something like f(x) = [linear equation in x], where the independent variable x acts like a counter because we restrict its values to {1, 2, 3, ...}.
The domain of this function is the set of natural numbers; the range is the set of numbers in the sequence.
The function associates 1 with -12, 2 with -5, 3 with 2, ...
You write the function's algebraic definition by using the first number in the sequence, the variable x, and a value that you get from examining the common difference between the numbers in the sequence. You can find the steps from the Google search.
If you're familiar with linear functions and finding equations of lines, then you may already know how to find an algebraic definition for the numbers in this sequence without realizing it.
You can think of the numbers in the sequence as y-coordinates.
Plotting the points from an arithmetic sequence results in points that are co-linear on the graph. In other words, the graph of some linear function goes through them.
(x, y)
(1, -12)
(2, -5)
(3, 2)
(4, 9)
(5, 16)
.
.
.
You only need two points to determine a linear equation.
Use the coordinates from the first two points to determine the slope. Use the resulting slope and the coordinates from the first point to write an equation in point-slope form, and then solve for y to get slope-intercept form.
Then switch back to function notation by replacing the symbol y with the symbol f(x), and make sure you explicitly state the restricted domain by writing something like "for x such that x is a natural number" or "x = {1, 2, 3, ...}".
This exercise is a good way to think about the slope of a line. It is the rate at which the y-coordinate increases for each increase in x. There is a direct relationship between this rate (slope) and the common difference between the terms in the sequence. Or, said another way, as you go from one number in the sequence to the next, the rate (slope) tells you how much you need to increase that first number in order to get the next one.
After reading up on arithmetic sequences, come back if you need more help. Please show any work that you're able to do, and try to say something about why you're stuck.
Cheers,
~ Mark
You could also try "arithmetic sequence progression" as keywords to get even more relevant hits.
Your function will be something like f(x) = [linear equation in x], where the independent variable x acts like a counter because we restrict its values to {1, 2, 3, ...}.
The domain of this function is the set of natural numbers; the range is the set of numbers in the sequence.
The function associates 1 with -12, 2 with -5, 3 with 2, ...
You write the function's algebraic definition by using the first number in the sequence, the variable x, and a value that you get from examining the common difference between the numbers in the sequence. You can find the steps from the Google search.
If you're familiar with linear functions and finding equations of lines, then you may already know how to find an algebraic definition for the numbers in this sequence without realizing it.
You can think of the numbers in the sequence as y-coordinates.
Plotting the points from an arithmetic sequence results in points that are co-linear on the graph. In other words, the graph of some linear function goes through them.
(x, y)
(1, -12)
(2, -5)
(3, 2)
(4, 9)
(5, 16)
.
.
.
You only need two points to determine a linear equation.
Use the coordinates from the first two points to determine the slope. Use the resulting slope and the coordinates from the first point to write an equation in point-slope form, and then solve for y to get slope-intercept form.
Then switch back to function notation by replacing the symbol y with the symbol f(x), and make sure you explicitly state the restricted domain by writing something like "for x such that x is a natural number" or "x = {1, 2, 3, ...}".
This exercise is a good way to think about the slope of a line. It is the rate at which the y-coordinate increases for each increase in x. There is a direct relationship between this rate (slope) and the common difference between the terms in the sequence. Or, said another way, as you go from one number in the sequence to the next, the rate (slope) tells you how much you need to increase that first number in order to get the next one.
After reading up on arithmetic sequences, come back if you need more help. Please show any work that you're able to do, and try to say something about why you're stuck.
Cheers,
~ Mark