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Suppose that we have an equation y = ax^2 + bx + c whose graph is a parabola with vertex (-4,7), vertical axis of symmetry, and contains the point (2,-1).  What is (a,b,c)?

 Sep 22, 2020
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Suppose that we have an equation y = ax^2 + bx + c whose graph is a parabola with vertex (-4,7), vertical axis of symmetry, and contains the point (2,-1).  What is (a,b,c)?

 

Hello Guest!

 

\(V (-4,7)\\ P_1(2,-1)\)

Because of the symmetry around \( x = -4\), is

\(P_2(-10,-1)\)

The coordinates of the 3 points are used in equation \(y = ax^2 + bx + c\), and the resulting equations are solved for a, b, c.

 

I. \(7=16a-4b+c\)

II. \(-1=4a+2b+c\)

III. \(-1=100a-10b+c\)

 

III. - II. \(0=96a-12b\)

I. - II. \(8=12a-6b\)

 

III. - II.           \(0=96a-12b\)

- 2 * (I. - II.) \(\underline{16=24a-12b}\)

                \(-16=72a\)

                      \(a=-\frac{2}{9}\)

 III. - II.          \(0=96\cdot (-\frac{2}{9})-12b\\ 0=-21.\overline{3}-12b\)

                      \(b=-1.\overline{7}\)

III.              \(-1=100a-10b+c\)

                  \(-1=-100\cdot \frac{2}{9} +10\cdot 1.7+c\)

                      \(c=3.\overline{4}\)

                      laugh  !


 

 Sep 22, 2020
edited by asinus  Sep 22, 2020

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