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

Verified step by step guidance

1. 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: \(y^2 - 2y = -12x + 35\).
2. 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: \(y^2 - 2y + 1 = -12x + 35 + 1\).
3. 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 \((y - 1)^2 = -12(x - 3)\).
4. 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 \((3, 1)\).
5. 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$.

Key Concepts

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.