Question

The vertex of this parabola is at (2, -1). When the y-value is 0, the x-value is 5. What is the coefficient of the squared term in the parabola's equation?

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Answer 1

hint:
here we have to start from this equation:
\[x = a{y^2} + by + c\]
and we have to find the values of the coefficients \(a,b,c\)

Answer 2

Now I substitute \(x=5,y=0\), so we get:
\[5 = a \cdot {0^2} + b \cdot 0 + c \Rightarrow c = 5\]

Answer 3

from the general theory, the y-coordinate \(y_V\) of the vertex, is:
\[{y_V} = \frac{{ - b}}{{2a}}\]

Answer 4

again, susbstituting \(y=-1\), we get:
\[{y_V} = \frac{{ - b}}{{2a}} = - 1 \Rightarrow b = 2a\]

Answer 5

finally, I substitute \(x=2\) when \(y=-1\), so I get:
\[2 = a - b + c\]

Answer 6

then we get the subsequent linear system:
\[\left\{ \begin{gathered}
2 = a - b + c \hfill \\
b = 2a \hfill \\
c = 5 \hfill \\
\end{gathered} \right.\]
please solve such system for \(a,b,c\)