If f(x)=ax^2+bx+c is a quadratic function for which f(0)= -4, f(6) = -6, and f(9) = 2, find coefficients a, b, and c.

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Yahir Wolfe · Accepted Answer

For this problem notice that there are 3 unknowns we are looking for, a,b, and c (the coefficients of the quadratic equation). However, we are given information that tells us the quadratic equation must satisfy 3 conditions. Using these 3 conditions, we can set up a system to find the 3 unknowns.First, plug in each of the given conditions into the general quadratic equation, f(x)=ax2+bx+c(1) f(0)=-4f(0)=a(0)2+b(0)+c=0+0+c=c=−4c=−4f(6)=a(6)2+b(6)+c=36a+6b+c=−636a+6b+c=−6f(9)=a(9)2+b(9)+c=81a+9b+c=281a+9b+c=281a+9b+c=236a+6b+c=%u22126 36a+6b+c=−6c=%u22124 c=−4We can substitute c=-4 into the first two equations to get81a+9b−4=236a+6b−4=−6Simplifying(1) 81a+9b=6(2) 36a+6b=−2Solving equation (1) for a,a=6−9b81Substituting a into equation (2),36(6−9b81)+6b=−2Solving for b,b=−73Substitute b back into a to solve for a,a=6−9(−7/3)81=13So, a=13,b=−73.c=−4 and you get the final quadratic equationf(x)=13x2−73x−4

aftredingzo · Accepted Answer

f(0) = c = -4f(6) = 36a+6b-4 = -6 -&amp;gt; 18a+3b = -1f(9) = 81a+9b-4 = 2 -&amp;gt; 27a + 3b = 2(27a-18a) = 2+1 -&amp;gt; a = 1/3 -&amp;gt; b = (-1-6)/3 = -7/3

If f(x)=ax^2+bx+c is a quadratic function for which f(0)= -4, f(6) = -6, and f(9) = 2, find coefficients a, b, and c.

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