|x |y |
|1 |1 |
|2 |4 |
|3 |9 |
|4 |16 |
In the table on the left, I observe that from the y value 1 to the y value 4 there is an increase of 3. From the y values 4 to 9, there is an increase of 5. From the y values 9 to 16, there is an increase of 7. This shows that it goes: +3, +5, +7, which is then increasing by 2 between each of those numbers.
Below, is the graph of y=x2
The equation y=x2 comes from the general equation y= ax2-bx+c. Y=x2 is the same as y=x2+0x+0. Therefore, a=1, b=0, and c=0.
The next example I am going to show you is similar to the one above. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1, 3, 6, 10, 15, 21, 28, 36).
|Tn |y | |1 |1 | |2 |3 | |3 |6 | |4 |10 | |5 |15 | |6 |21 | |7 |28 | |8 |36 |
In the table on the left, I observe that from the y value 1 to the y value 3 there is an increase of 2. From the y values 3 to 6, there is an increase of 3. From the y values 6 to 10, there is an increase of 4. This shows that it goes: +2, +3, +4, which is then increasing by 1 between each of those numbers.
This follows the form of an arithmetic sequence, because each new term is equal to the sum of the previous term plus the term number. A formula for this sequence can then be derived form the formula of an arithmetic sequence, shown below. n represents the term number, a represents the number of dots in the first term, and d represents the common difference between the terms.
Since the first team has one dot, this means that for this sequence, a=1. The difference between each term is represented by d. Therefore, d=1. By substituting these values into the arithmetic sequence formula, I get the equation: [pic]
Which is then simplified to:
This formula above can be used to calculate the number of dots of any triangular number is the sequence, n being greater than 0.
Graphing the points from the table above, gives me this graph:
is correct because the points in the table fall exactly on the graphed line.
Above, the equation: [pic] also follows the general equation y=ax2+bx+c. If I move around the equation to fit the general form, [pic]. Therefore, a= ½, b= ½, and c=0.
Next, I will be considering stellar shapes with p vertices, leading to p-stellar numbers. I will be showing you stages S1-S6 in diagrams below.
|Stage number |Number of dots |Dot increase between stages | |S1 |1 | | |S2 |13 |+12 | |S3 |37...