Investigate the link between Pascal's triangle and 2 and its powers

Outside of probability, Pascal's Triangle is also used for: Algebra, where coefficient of polynomials can be used to find the numbers in Pascal's triangle.  Pascal's Triangle is an arithmetical triangle you can use for some neat things in mathematics.

The entries in Pascal's triangle are actually the number of combinations of N take n where N is the row number starting with N = 0 for the top row and n is the nth number in the row counting from left to right where the n = 0 number is the first number.

The mathematical formula for the number of combinations without repetition is N!/(n!(N-n)!).

Step-by-step explanation:

To construct Pascal's triangle, start with a 1. Then, in the next row, write a 1 and 1. It's good to have spacing between the numbers. In the third row, we have 1 and 1 on the outside slopes. The 2 comes from adding the two numbers above and adjacent. Thus, we are adding the number on the left, 1, with the number on the right, 1, to get 1 + 1 = 2.

In the next row, the 3 comes from adding the 1 and the 2. This particular Pascal's triangle stopped at 1 5 10 10 5 1, but we could have continued indefinitely.

given that triangle abc is transformed two times to get triangle

a"b"c"

first abc is transformed in to a'b'c' as follows:

the coordinates of a are (1,-1) and transformed into a'(-1,1)

similarly b (4,-2) became b'(-4,2) and

c(7,-2) became c'(-7,2)

i.e. (x,y) becomes (-x,-y)

this is nothing but reflection about a point here origin.

thus first transformation is reflection on the origin.

next is exactly shifting 3 units down

source:

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1