equation for this table
x= 2, 3, 4, 5
y= 6, 4, 3, 2.4
Is it a straight line?
m1 = (6 - 4) / (2 - 3) = 2 / -1 = -2
m2 = (4 - 3) / (3 - 4) = 1 / -1 = -1
m3 = (3 - 2.4) / (4 - 5) = 0.6 / -1 = -0.6
The slope keeps changing, so this is not a straight line.
Step up to the next level.
y = ax^2 + bx + c
At any point, we have:
6 = a(2)^2 + b2 + c
1) 6 = 4a + 2b + c
4 = a(3)^2 + b3 + c
2) 4 = 9a + 3b + c
3 = a(4)^2 + b4 + c
3) 3 = 16a + 4b + c
Solve that system of equations for a, b and c.
Multiply equation 1 by 4.
4 * 6 = 4 * (4a + 2b + c)
4) 24 = 16a + 8b + 4c
Subtract equation 3 from equation 4.
24 = 16a + 8b + 4c
-(3 = 16a + 4b + c)
--------------------------
21 = 4b + 3c
5) 21 = 4b + 3c
Multiply equation 1 by 9.
9 * 6 = 9 * (4a + 2b + c)
6) 54 = 36a + 18b + 9c
Multiply equation 2 by 4.
4 * 4 = 4 * (9a + 3b + c)
7) 16 = 36a + 12b + 4c
Subtract equation 7 from equation 6.
54 = 36a + 18b + 9c
-(16 = 36a + 12b + 4c)
-----------------------------
38 = 6b + 5c
8) 38 = 6b + 5c
We have eliminated a in two equations. Now,
we will eliminate b.
Multiply equation 5 by 3.
3 * 21 = 3 * (4b + 3c)
9) 63 = 12b + 9c
Multiply equation 9 by 2.
2 * 38 = 2 * (6b + 5c)
10) 76 = 12b + 10c
Subtract equation 9 from equation 10.
76 = 12b + 10c
-(63 = 12b + 9c)
-----------------------
13 = c
c = 13 <<<<<<<<<<<<<<<<<<<
We have several equations with only b and c: 5, 8, 9 and 10.
Let's use equation 5.
21 = 4b + 3c
21 = 4b + 3(13)
21 = 4b + 39
21 - 39 = 4b + 39 - 39
-18 = 4b
-18 / 4 = 4b / 4
b = -4.5 <<<<<<<<<<<<<<<<<<<
Using equation one, we can solve for a.
6 = 4a + 2b + c
6 = 4a + 2(-4.5) + 13
6 = 4a - 9 + 13
6 = 4a + 4
6 - 4 = 4a + 4 - 4
2 = 4a
2 / 4 = 4a / 4
a = 1/2 <<<<<<<<<<<<<<<<<<<
We have found a, b and c, the constants in the
second degree equations that gave us equations 1, 2 and 3.
6 = ax^2 + bx + c
6 = a(2)^2 + b2 + c
6 = 1/2(2)^2 + (-4.5)(2) + 13
6 = 1/2(4) - 9 + 13
6 = 2 - 9 + 13
6 = 15 - 9
6 = 6
The constants satisfied equation 1.
4 = ax^2 + bx + c
4 = 1/2(3)^2 + (-4.5)3 + 13
4 = 1/2(9) + (-13.5) + 13
4 = 4.5 - 13.5 + 13
4 = 4.5 - 0.5
4 = 4
The constants satisfied equation 2.
3 = ax^2 + bx + c
3 = 1/2(4)^2 + (-4.5)4 + 13
3 = 1/2(16) + (-4.5)4 + 13
3 = 8 - 18 + 13
3 = 8 - 5
3 = 3
The constants satisfied equation 3.
The equation we are looking for is:
y = 1/2x^2 - 4.5x + 13