Modelling a quadratic - Systems of linear equations

Ari

Sep 2007
50
1
Hi, i cant figure out how to set up this question:
ax^2 + bx + c passes through (-1,5) and has a horizontal tangent at (1,7) find the coefficents a b and c

Thanks.
 
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dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Hi, i cant figure out how to set up this question:
ax^2 + bx + c passes through (-1,5) and has a horizontal tangent at (1,7) find the coefficents a b and c

Thanks.
\(\displaystyle a(-1)^2-1b+c=5\rightarrow a-b+c=5\)

derivative

\(\displaystyle 2xa+b=0\) when x=-1
 
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Ari

Sep 2007
50
1
k im confused, how does that help me find the the values for a,b,c?

edit: and can you please explain how you got that? i feel very confused atm lol.
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Setup the coefficient matrix for the two equations and solve simultaneously.
 

Ari

Sep 2007
50
1
Setup the coefficient matrix for the two equations and solve simultaneously.
i just started this course... which two equations are you talking about?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
i just started this course... which two equations are you talking about?
1: \(\displaystyle a-b+c=5\)
2: \(\displaystyle -2a+b=0\)
 
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Ari

Sep 2007
50
1
1: \(\displaystyle a-b+c=5\)
2: \(\displaystyle 2a+b=0\)
oh okay, how did you get that second equations from the

\(\displaystyle a-b+c=5\)\

edit: nevermind i see it now. its from the originial one.

now how exactly do i solve them simultaneously? i can get the parametric solutions, but how do i get the number values?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
\(\displaystyle a(-1)^2-1b+c=5\rightarrow a-b+c=5\)

derivative

\(\displaystyle 2xa+b=0\rightarrow -2a+b=0\) when x=-1
The 2nd equation came from the original equation.
 
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Ari

Sep 2007
50
1
is the answer just going to be a,b,c in parametric form or will there be coefficent values, if so how do i get values for them?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
is the answer just going to be a,b,c in parametric form or will there be coefficent values, if so how do i get values for them?
Reduced row echelon form of the coefficient matrix.

\(\displaystyle \begin{bmatrix}
1 & -1 & 1 & 5\\
-2 & 1 & 0 & 0
\end{bmatrix}\)
 
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