Show that H can be modeled as a quadratic function of D

Question

There is a table. The distances (D) in inches are 0, 1, 2, 3, and 4.  The Depth (h) in inches are 1, 1.25, 2, 3.25, 5

Answer 1

Quadratic functions follow the form f(x) = Ax² + Bx + C, or if we're using D here, h(D) = AD² + BD + C, for some constants A, B, and C.  We can use the given points to try to find the constants.

 

First, try the (0,1) point:

 

1 = A(0)² + B(0) + C

1 = C

 

That's simple enough, and gives us one constant.  Fill it in, and try the (1,1.25) point

 

1.25 = A(1)² + B(1) + 1

.25 = A + B

 

Doesn't directly tell us A or B, but it gives us one relationship between them.  On to (2,2)

 

2 = A(2)² + B(2) + 1

1 = 4A + 2B

Another relationship - we can use this one with the last to find A and B.  First, double both sides of the earlier one:

 

.5 = 2A + 2B

 

Subtract this from the latter:

 

  (1 = 4A + 2B)

-(.5 = 2A + 2B)

-------------------

.5 = 2A

A = .25

 

Substitute this back into either of them:

.25 = .25 + B

B = 0

 

The first three points thus tell us our function:  h(D) = .25D² + 1

Now we can verify that it works with the last two points.

 

h(3) = .25*9+1 = 2.25 + 1 = 3.25, which is what we're supposed to get.

h(4) = .25*16 + 1 = 4 + 1 = 5, which is what we're supposed to get.  The equation works.