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Which of the following is the equation of the parabola described with vertex at $(5, -3)$ , axis parallel to the $y$ -axis and passing through the point $(1, 1)$ ?

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Write equations of parabolas in vertex form using properties

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Q. Which of the following is the equation of the parabola described with vertex at

(5, -3)

, axis parallel to the

y

-axis and passing through the point

(1, 1)

?

Identify Vertex Form: Identify the vertex form of a parabola with a vertical axis. $\newline$ The vertex form is $y = a(x-h)^2 + k$ where (h, k) is the vertex. $\newline$ Here, h = $5$ and k = $-3$ .
Substitute Vertex: Substitute the vertex into the vertex form equation. $\newline$ Using h = $5$ and k = $-3$ , the equation becomes $y = a(x-5)^2 - 3$ .
Find Value of a: Use the point $(1, 1)$ to find the value of $a$ . $\newline$ Substitute $x = 1$ and $y = 1$ into the equation: $\newline$ $1 = a(1-5)^2 - 3$ $\newline$ $1 = a(16) - 3$ $\newline$ $4 = 16a$ $\newline$ $a = \frac{4}{16}$ $\newline$ $a = \frac{1}{4}$
Write Final Equation: Write the final equation of the parabola. $\newline$ Substitute $a = \frac{1}{4}$ back into the equation: $\newline$ $y = \left(\frac{1}{4}\right)(x-5)^2 - 3$

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