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. x² - 2x - 4y + 9 =0

Verified step by step guidance

1. Start with the given equation: \(x^2 - 2x - 4y + 9 = 0\).
2. Group the \(x\) terms together and move the \(y\) term and constant to the other side: \(x^2 - 2x = 4y - 9\).
3. Complete the square for the \(x\) terms. Take half of the coefficient of \(x\) (which is \(-2\)), square it, and add it to both sides: half of \(-2\) is \(-1\), and \((-1)^2 = 1\). So, add \$1\( to both sides: \)x^2 - 2x + 1 = 4y - 9 + 1$.
4. Rewrite the left side as a perfect square: \((x - 1)^2 = 4y - 8\). Then isolate \(y\): \((x - 1)^2 = 4(y - 2)\).
5. Identify the vertex from the standard form \((x - h)^2 = 4p(y - k)\), where the vertex is at \((h, k)\). Here, \(h = 1\) and \(k = 2\). Use the value of \$4p\( to find \)p\(, which helps determine the focus and directrix. The focus is at \)(h, k + p)\( and the directrix is the line \)y = k - p$.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)^2 = k, which helps convert equations into standard form. This technique involves adding and subtracting a constant to create a perfect square trinomial, making it easier to identify key features of conic sections like parabolas.

Standard Form of a Parabola

The standard form of a parabola with a vertical axis is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p determines the distance to the focus and directrix. Converting to this form allows for straightforward identification of the vertex, focus, and directrix, which are essential for graphing the parabola.

Vertex, Focus, and Directrix of a Parabola

The vertex is the parabola's turning point, the focus is a fixed point inside the curve, and the directrix is a line outside the curve. The parabola is the set of points equidistant from the focus and directrix. Knowing these elements helps in accurately graphing and understanding the parabola's shape and orientation.