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
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: $\displaystyle y=mx+c$ (which would make it a line)?
We have: $\displaystyle (1,6)$
Substituting this into $\displaystyle y=mx+c$ gives:
1) $\displaystyle 6=m+c$
We have $\displaystyle (2,15)$
Substituting again:
2)$\displaystyle 15=2m+c$
Solving simultaneously:
2)-1)
$\displaystyle 9=m$ so
$\displaystyle c=-3$
$\displaystyle y=9x-3$ is the equation of the line running through these two points. Does $\displaystyle (3,27)$ lie on this line?
Test:
$\displaystyle 27\neq~9(3)-3$
So it isn't a line.
Could it be something of the form $\displaystyle y=ax^2+bx+c$? You could use a similar test.
What is the whole question? Find a random curve going through these 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)
Case 1) $\displaystyle y=1$ then$\displaystyle x=6$.Let the number of hexagons be $\displaystyle y$ and the number of sides( straws) be $\displaystyle x$.
Hexagon has 6 sides.
Case 2) $\displaystyle y=3$ then $\displaystyle x=15$
Note That: 3 hexagons would mean $\displaystyle 3 \times 6 = 18 $sides but there are $\displaystyle 15$ sides. This means that some sides are common.
Case 3) $\displaystyle y=6$ then $\displaystyle x=27$Common Sides = 3
We can establish a relation between x and y that:$\displaystyle x=6y-C$, where $\displaystyle C$ is the number of common sides.Common Sides = 9
To relate x,y and C, Let us plot down their values in the form of a table.
and so on...
You will observe that $\displaystyle C$ is increasing in arithmetic progression (Every time we add 1 hexagon, $\displaystyle C$ increases by 2).$\displaystyle Note That:$These values have been obtained by drawing the appropriate no. of hexagons.
Therefore:
$\displaystyle \\a_n=a+(n-1)d \\ C_y=1+(y-2)2 \\ C_y=2y-3$
So: $\displaystyle x=6y-(2y-3)=4y+3$
The relation between x and y is $\displaystyle x=4y+3$.
$\displaystyle Note That:$This relation is valid for all values except y=1.