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${\text{Write an equation of a horizontal parabola with a vertex of }}\left( {1, - 2} \right){\text{ and passing }} \\ {\text{through }}\left( {9, - 4} \right). \\$
${\text{Let the equation of the horizontal parabola is }}{\left( {y - h} \right)^2} = 4a\left( {x - k} \right) \\ {\text{because it is passing through}}\left( {9, - 4} \right)\;{\text{which is right side of the parabola }} \\ {\text{i}}{\text{.e parabola is opening right side}}{\text{.}} \\ {\text{so the vertex of the parabola is }}\left( {k,h} \right) \\ {\text{So, compare the vertex from the given vertex}} \\ {\text{i}}{\text{.e }}\left( {1, - 2} \right) \\ \Rightarrow k = 1,\;h = - 2 \\ {\text{So, the equation of parabola is}} \\ {\left( {y + 2} \right)^2} = 4a\left( {x - 1} \right) \\ {\text{Now it is passing through }}\left( {9, - 4} \right){\text{ which satisfy the parabola}} \\ \Rightarrow {\left( { - 4 + 2} \right)^2} = 4a\left( {9 - 1} \right) \\ \Rightarrow 4 = 4a \times 8 \Rightarrow a = \dfrac{1}{8} \Rightarrow 4a = \dfrac{1}{2} \\ \Rightarrow {\text{Equation of parabola is}} \\ {\left( {y + 2} \right)^2} = \dfrac{1}{2}\left( {x - 1} \right) \\ {\text{So, this is your required equation}}{\text{.}} \\ {\text{NOTE: - In this type of problem first check from given condition parabola is opening which side,}} \\ {\text{ then satisfying the condition according to given problem statement you will get your answer}}{\text{.}} \\$