Function approximation

Question

The function

 

g(x)=|x-2|+|x-1.5|+|x-1|+|x-0.5|+|x|+|x+0.5|+|x+1|+|x+1.5|+|x+2|

 

can be approximated on the interval [-3,3] by the quadratic function y=ax^2+b, where a and b are both integers. Find a and b.

Answer 1

Hi Cassandra,

 

First of all I am going to use q(x) instead of g(x), you will see why later.

 

When x=0, q(0) = 10 which is a minimum value for q. That will be the vertex of the approximating parabola. 

 

Since we are looking for a parabola in the form y = ax²+b, when x=0 we want y=10, which results in b=10.

 

How to determine a? Not sure this is what was intended but this is my guess. It would be great if the parabola went through the points on the graph of q nearest the "vertex". Those would correspond to the x-values -0.5 and 0.5.

 

Since q(0.5) = 10.5 (similarly for q(-0.5), we want the equation of the parabola to satisfy:

 

10.5 = a(0.5)²+10

 

.25a = .5

 

a = 2

 

See this graph to get a better idea of what is going on:

 

https://www.desmos.com/calculator/harnrbseda