Writing equations from patterns?

Question

I need help with writing equations from patterns. I can do it when there is a common pattern and ratio but I don't know how to write equations for ones that don't. Could you please tell me the equations for the following patterns and then explain how you arrived at the answers?

1. {(2,12),(1,48),(-1,48),(-2,12),(-4,3)}
2. {(-3,10),(-2,5),(-1,2),(0,1),(1,2),(2,5),(3,10)
3. {(6,-4),(2,-12),(1,-24),(-3,8),(-6,4)}

Thankyou very much!!!!

The patterns got cut off above, so here they are again:

1.[(2,12),(1,48),(-1,48),(-2,12),(-4,3)]
2. [(-3,10),(-2,5),(-1,2),(0,1),(1,2),(2,5),(3,10)]
3. [(6,-4),(2,-12),(1,-24),(-3,8),(-6,4)]

Answer 1

What kind of equations are you learning about? Are you learning about relations that don't fit equations?
In general, if you have a linear relation, you'll have the same difference between sucessive terms. For example, the first number might be going up by 1 each time, and the second number going down by 3 each time. But none of these look like that.
These also look like the could be points on a parabola so let's explore this some more.
If they're points on a parabola they'd fit an equation of the form
y = a(x+h)^2 + k
or maybe we can simplify this further and try equations of the form y = ax^2 + k
This looks promising, as the second set of numbers appears to fit the form y = x^2 + 1
Try squaring the first number and seeing what you need to do with it to generate the second number. See if there is a pattern.
The first one looks like it might be a parabola that faces downward. This would be something of the form y = c - a(x-h)^2
Try graphing these numbers and see if you can figure out what c and a might be. C would be the number where it peaks out on the the y axis, at the vertex.
The third one is a little harder to get hold of, the (-3, 8) seems to break all the rules. Looks like it might be a cubic. Again, plot the points, and experiemnt with relations of the form y = ax^3 + c

Answer 2

I'd recommend sketching the points and see what they suggest. If you like to be 'high tech' then you could enter them into a list in your TI-83 and plot them, also looking for a pattern.

The other, even more basic, thing to do is examine the points looking for features.

For example: symmmetery
for
1. those 4 points are symmetrical about the y axis, with the y(x=1) > y(x=2), this suggests a parabola opening down.

Still, a simple plot is easy and effective.

Answer 3

its easy!!!! suppose u have the pattern +1^2, 2^2, 3^2, 4^2,... that is square of the first term,square of second term, square of third term and so on let n be any term in this sequence then the nth term in this sequence will be??? umm umm?? right its (n)^2 suppose u have a series as shown below 2,5,8,11..... well to write the nth term in such cases u need to understand that it is an AP. that is arithmetic profression(as the difference between cosecutive numbers is same) i.e. 3 in such cases the nth term is given by nth ter,m = a+(n-1)d where a = first term f the sequence ,n = the no. of term u need and d is the commom differene(that is the difference between consecutive numbers! u say that u end up relating to the previous number..well it happens in the starting but a lil practise will make u a booommmm!! hah! just try it! practise it its simple, interesting and i love it!!!!!have a nice day:D

Answer 4

Put spaces in your patterns so they don't get cut off. Sometimes if you graph the coordinates as points, it's easier to see the pattern.