A quadratic pattern has a third term equal to 2,a fourth term equal to -2 and the sixth term equal to-16 . calculate the second difference of this quadratic pattern show all the working
make a chart
x y ∆y ∆(∆y)
1
2
3 2
4 -2 -4
5 k (k+2)
6 -16 (-16-k)
but we know that the first differences for a quadratic
form an arithmetic sequence, and the 2nd difference is a constant
so (-16-k) - (k+2) = k+2 + 4
-3k = 24
k = - 8
I can now complete the chart:
x y ∆y ∆(∆y)
1
2
3 2
4 -2 -4 --- -2
5 -8 -6 --- -2
6 -16 -8 -- -2
and I would be able to complete the entire chart without
actually finding the quadratic.
x y ∆y ∆(∆y)
1 4 2 ----- -2
2 4 0 ----- -2
3 2 -2 ---- -2
4 -2 -4 --- -2
5 -8 -6 --- -2
6 -16 -8 -- -2
or:
you could find the actual quadratic ax^2 + bx + c, by
setting up 3 equations in 3 variables, then filling in the missing parts.
To find the second difference of a quadratic pattern, we first need to find the first difference and then the second difference.
Let's start by finding the first difference:
The third term is 2, and the fourth term is -2. Hence, the difference between these two terms is:
-2 - 2 = -4
The fourth term is -2, and the sixth term is -16. The difference between these two terms is:
-16 - (-2) = -14
The first difference is the difference between consecutive terms. Now let's find the second difference:
The first difference between the third and fourth terms is -4, and the first difference between the fourth and sixth terms is -14. Therefore, the second difference is the difference between these two first differences:
-14 - (-4) = -14 + 4 = -10
So, the second difference of this quadratic pattern is -10.
To find the second difference of a quadratic pattern, we first need to determine the general formula for the sequence. Since it is a quadratic pattern, the formula will be of the form:
f(n) = an^2 + bn + c
where f(n) represents the nth term of the sequence, and a, b, and c are constants that need to be determined.
To find the values of a, b, and c, we can substitute the given terms of the sequence into the equation.
The third term is equal to 2, so we have:
f(3) = a(3)^2 + b(3) + c = 2
Simplifying this equation, we get:
9a + 3b + c = 2
The fourth term is -2, so we have:
f(4) = a(4)^2 + b(4) + c = -2
Simplifying this equation, we get:
16a + 4b + c = -2
Finally, the sixth term is -16, so we have:
f(6) = a(6)^2 + b(6) + c = -16
Simplifying this equation, we get:
36a + 6b + c = -16
Now, we have a system of three equations with three unknowns:
9a + 3b + c = 2
16a + 4b + c = -2
36a + 6b + c = -16
Solving this system of equations, we find:
a = -2
b = -9
c = 28
Now that we have determined the values of a, b, and c, we can rewrite the formula for the quadratic pattern as:
f(n) = -2n^2 - 9n + 28
To find the second difference, we need to calculate the difference between consecutive terms and then find the difference between those differences.
For example, let's calculate the differences between terms:
1st difference = f(2) - f(1) = ( -2(2)^2 - 9(2) + 28) - ( -2(1)^2 - 9(1) + 28) = -1
2nd difference = f(3) - f(2) = ( -2(3)^2 - 9(3) + 28) - ( -2(2)^2 - 9(2) + 28) = -4
Therefore, the second difference of this quadratic pattern is -4.