Consider the points (-1, 1), (1,-5), and (3,-3). (a) Find an equation for the pa

Question

Consider the points (-1, 1), (1,-5), and (3,-3).
(a) Find an equation for the parabola that passes through the points.
(b) Find the least squares line that best ts these points.

Explanation / Answer

so we all know that the equation of parabola's equation is y=ax^2+bx+c Therefore we plug the points given in this formula and we'll have 1=a-b+c -5=a+b+c 3=9a+3b+c So the goal is to find a,b,c, this can be abit tricky since there is no value with y=0 so we have to use double elimination at first we take the first and second equation and eliminate one variable, and then we take the first and third equation and eliminate the same variable 1=a-b+c -5=a+b+c ________+ -4=2a+2c------> -2=a+c 1=a-b+c multiplied by three ---> 3=3a-3b+3c 3=3a-3b+3c 3=9a+3b+c _____________+ 6=12 a+6c Divided by 6---> 1=2a+c So we once again do an elimination to get a and b value a+c=-2 2a+c=1 ________- -a=-3 a=3 and b=-3 plug them in one of the equations you will find c=-5 therefore the equation of parabola will be y=3x^2-3b-5

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