+14913 what the descride transformation (3,1),(0,-5),(-4,-2)
A graph through the three points may be a circle or a second-degree parabola, 3rd degree or higher degree.
Circle:
\(x_M=\frac{x_1+x_2+x_3}{3}=\frac{3+0-4}{3}\\x_M=\frac{7}{3}\)
\(y_M=\frac{y_1+y_2+y_3}{3}=\frac{1-5-2}{3}\\y_M=-2\)
\(r=\sqrt{(x_M-x_1)^2+(y_M-y_1)^2}\)
\(r=\sqrt{(\frac{7}{3}-3)^2+(-2-1)^2}\)
\(r=\sqrt{(-\frac{2}{3})^2+(-3)^2}=\sqrt{\frac{4}{9}+9}=\sqrt{\frac{85}{9}}\)
\(r=\frac{\sqrt{85}}{3}\)
Circular function
\(x^2+y^2=r^2\)
\(x^2+y^2=\frac{85}{9}\)
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+14913 transformation (3,1),(0,-5),(-4,-2)
Parable of three points
\(y=ax^2+bx+c\)
A ) \(1=a\cdot 3^2+3b+c\\9a+3b+c=1\)
B) \(-5=0a+0b+c\\c=-5\)
C) \(-2=16a-4b+c\\16a-4b-5=-2\)
A) \(9a+3b-5=1\\a=\frac{6-3b}{9}\)
C) \(16a-4b=3\\a=\frac{4b+3}{16}\)
A)&C) \(\frac{6-3b}{9}=\frac{4b+3}{16}\\96-48b=36b+27\\84b=69\\b=\frac{69}{84}=\frac{3\cdot23}{2\cdot2\cdot21}\\b=0.821\)
A)&C)⇒A) \(a=\frac{6-3\cdot \frac{69}{84}}{9}=\frac{6}{9}-\frac{3\cdot69}{9\cdot84}\\a=0.393\)
\(\color{blue}{y=ax^2+bx+c\\y=0.393x^2+0.821x-5}\)
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