In Exercises 43–48, 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. x^2 - 2x - 4y + 9 =0

Verified step by step guidance

1. Start by rearranging the given equation to group the x terms together: $x^2-2x=4y-9$.
2. Complete the square for the x terms. Take the coefficient of x, which is -2, divide it by 2 to get -1, and then square it to get 1. Add and subtract this square inside the equation: $x^2-2x+1-1=4y-9$.
3. Rewrite the equation with the completed square: $(x-1{)}^2-1=4y-9$.
4. Move the constant term to the other side to isolate the completed square: $(x-1{)}^2=4y-8$.
5. Divide the entire equation by 4 to express it in standard form: $(x-1{)}^2=4(y-2)$.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique allows us to rewrite the equation in a form that makes it easier to identify key features of the parabola, such as its vertex. By rearranging the equation and adjusting constants, we can express it in standard form, which is essential for further analysis.

Standard Form of a Parabola

The standard form of a parabola is typically expressed as (x - h)² = 4p(y - k) for vertical parabolas or (y - k)² = 4p(x - h) for horizontal parabolas. Here, (h, k) represents the vertex of the parabola, and 'p' indicates the distance from the vertex to the focus and the directrix. Understanding this form is crucial for identifying the parabola's geometric properties.

Vertex, Focus, and Directrix

The vertex of a parabola is the highest or lowest point, depending on its orientation. The focus is a point located along the axis of symmetry, where all lines drawn parallel to the axis reflect off the parabola. The directrix is a line perpendicular to the axis of symmetry, equidistant from the vertex as the focus. Together, these elements define the parabola's shape and position in the coordinate plane.