Finding the equation given 3 points

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November 11th 2011, 07:35 PM
pikachu26134

Finding the equation given 3 points

I have been given 3 points; (1,6) (2,15) (3,27)
And i have to find the equation of the line/curve, although I have spent a long time on it and feel as though I am overlooking something. Any help would be appreciated
November 11th 2011, 08:04 PM
Quacky

Re: Finding the equation given 3 points

I think you need to be more specific with the question: there could be any number of curves going through those points, surely. What sort of curve are you looking for?

Let's see, is it of the form: $y=mx+c$ (which would make it a line)?

We have: $(1,6)$

Substituting this into $y=mx+c$ gives:

1) $6=m+c$

We have $(2,15)$
Substituting again:

2) $15=2m+c$

Solving simultaneously:
2)-1)
$9=m$ so
$c=-3$

$y=9x-3$ is the equation of the line running through these two points. Does $(3,27)$ lie on this line?
Test:
$27\neq~9(3)-3$

So it isn't a line.

Could it be something of the form $y=ax^2+bx+c$ ? You could use a similar test.

What is the whole question? Find a random curve going through these points?
November 11th 2011, 08:08 PM
pikachu26134

Re: Finding the equation given 3 points

Give the rule for the number of straws that Slade and Jade would need for the nth design of tessellating hexagons.
In the first pattern, there is one hexagon (6 straws), in the 2nd there is 3 hexagons (15 straws) and in the 3rd one there is 6 hexagons (27 straws)
November 12th 2011, 03:47 AM
sbhatnagar

Re: Finding the equation given 3 points

Quote:

Let the number of hexagons be $y$ and the number of sides( straws) be $x$ .
Hexagon has 6 sides.

Case 1) $y=1$ then $x=6$ .

https://lh4.googleusercontent.com/-U...M/s512/one.png

Case 2) $y=3$ then $x=15$

Quote:

Note That: 3 hexagons would mean $3 \times 6 = 18$ sides but there are $15$ sides. This means that some sides are common.

https://lh4.googleusercontent.com/-2...s640/three.png

Quote:

Common Sides = 3

Case 3) $y=6$ then $x=27$

https://lh5.googleusercontent.com/-r...E/s512/six.png

Quote:

Common Sides = 9

We can establish a relation between x and y that: $x=6y-C$ , where $C$ is the number of common sides.

Quote:

To relate x,y and C, Let us plot down their values in the form of a table.

https://lh6.googleusercontent.com/-a...s512/table.PNG

and so on...

Quote:

$Note That:$ These values have been obtained by drawing the appropriate no. of hexagons.

You will observe that $C$ is increasing in arithmetic progression (Every time we add 1 hexagon, $C$ increases by 2).

Therefore:
$\\a_n=a+(n-1)d \\ C_y=1+(y-2)2 \\ C_y=2y-3$

So: $x=6y-(2y-3)=4y+3$

The relation between x and y is $x=4y+3$ .

Quote:

$Note That:$ This relation is valid for all values except y=1.
November 12th 2011, 09:02 AM
Quacky

Re: Finding the equation given 3 points

Yet another example of the importance of posting the entire question, rather than the parts you think are relevant.