A quadratic pattern has a second term equal to 1, the third term equal to -6 and the fifth term equal to -14. Calculate the second difference
Hence, or otherwise, calculate the first term of the pattern
let f(x) = ax^2 + bx + c
given:
f(2) = 4a + 2b + c = 1
f(3) = 9a + 3b + c = -6
f(5) = 25a + 5b + c = -14
subtract the 1st from the 2nd : 5a + b = -7
subtract the 2nd from the 3rd: 16a + 2b = -8 ===> 8a + b = -4
subtract these last two:
3a = 3 ====> a = 1
sub into 5a + b = -7
5 + b = -7 =====> b = -12
sub into the first:
4 - 24 + c = 1 ===> c = 21
f(x) = y = x^2 - 12x + 21
table of values:
x y ∆y ∆(∆y)
0 21
1 10 -11
2 1 -9 2
3 -6 -7 2
4 -11 -5 2
5 -14 -3 2 ====> the 2nd difference is constant at 2
My mind locked in on this way of doing it, there probably is
an easier way
To find the second difference in a quadratic pattern, we need to look at the differences between consecutive terms.
Given that the second term is 1, the third term is -6, and the fifth term is -14, we can calculate the first differences:
First difference between the second and third terms:
-6 - 1 = -7
First difference between the third and fifth terms:
-14 - (-6) = -8
Now, let's calculate the second differences by finding the difference between these first differences:
Second difference = -8 - (-7) = -8 + 7 = -1
Therefore, the second difference in this quadratic pattern is -1.
To find the second difference of a quadratic pattern, we need to examine the differences between consecutive terms and the differences between those differences.
Let's first list out the terms of the quadratic pattern:
First term = ?
Second term = 1
Third term = -6
Fourth term = ?
Fifth term = -14
Now, let's calculate the differences between consecutive terms:
Difference between 2nd and 1st term: 1 - ? = 1
Difference between 3rd and 2nd term: -6 - 1 = -7
Difference between 4th and 3rd term: ? - (-6) = ?
Difference between 5th and 4th term: -14 - ? = -14
Now, let's calculate the differences between these differences:
Difference between (-7) and 1: -7 - 1 = -8
Difference between ? and (-7): ? - (-7) = ?
Difference between -14 and ?: -14 - ? = ?
Since we are interested in the second difference, we need to find the difference between the differences we calculated in the previous step:
Second difference = ? - (-8) = ? + 8
So, to calculate the second difference, we need to find the missing terms so that we have a clear pattern.
Let's calculate the value of the 4th term by using the differences between consecutive terms:
Fourth term = (-6) + (-7) = -13
Now, let's calculate the difference between the 4th and 3rd term:
Difference between 4th and 3rd term: -13 - (-6) = -7
Next, let's calculate the difference between -14 and the 4th term:
Difference between -14 and -13: -14 - (-13) = -1
Finally, let's calculate the second difference:
Second difference = -1 - (-7) = -1 + 7 = 6
So, the second difference of the quadratic pattern is 6.