Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. y² - 2y + 12x - 35 = 0

검증된 단계별 안내

Start by rewriting the given equation: $y^2 - 2y + 12x - 35 = 0$. Group the $y$ terms together and move the $x$ term and constant to the other side: $y^2 - 2y = -12x + 35$.

Complete the square for the $y$ terms. Take half of the coefficient of $y$, which is $-2$, divide by 2 to get $-1$, then square it to get $1$. Add $1$ to both sides to maintain equality: $y^2 - 2y + 1 = -12x + 35 + 1$.

Rewrite the left side as a perfect square: $(y - 1)^2 = -12x + 36$. Then isolate $x$ by moving terms: $(y - 1)^2 = -12x + 36$ becomes $(y - 1)^2 = -12(x - 3)$.

Identify the vertex from the equation in standard form $(y - k)^2 = 4p(x - h)$, where the vertex is at $(h, k)$. Here, the vertex is at $(3, 1)$.

Determine the value of $p$ by comparing $-12$ to \$4p$, so \(4p = -12$ which gives \)p = -3$. Use $p$ to find the focus at $(h + p, k)$ and the directrix as the vertical line $x = h - p$.

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주요 개념

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Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)^2 or (y - k)^2 by adding and subtracting terms. This technique helps convert equations into standard form, making it easier to identify key features like the vertex of a parabola.

Standard Form of a Parabola

The standard form of a parabola's equation reveals its geometric properties clearly. For parabolas opening horizontally or vertically, the equation is written as (y - k)^2 = 4p(x - h) or (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p relates to the distance between the vertex and the focus.

Vertex, Focus, and Directrix of a Parabola

The vertex is the parabola's turning point, the focus is a fixed point inside the parabola used to define it, and the directrix is a line perpendicular to the axis of symmetry. Knowing these helps in graphing the parabola and understanding its shape and orientation.

Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. y2 - 2y + 12x - 35 = 0