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 + 6x - 4y + 1 = 0

검증된 단계별 안내

Start by rewriting the given equation \(x^2 + 6x - 4y + 1 = 0\) to isolate the \(y\) terms on one side. Move all terms except those involving \(y\) to the other side: \(-4y = -x^2 - 6x - 1\).

Divide both sides of the equation by \(-4\) to solve for \(y\): \(y = \frac{1}{4}x^2 + \frac{3}{2}x + \frac{1}{4}\).

To complete the square for the \(x\) terms, focus on the quadratic expression \(\frac{1}{4}x^2 + \frac{3}{2}x\). Factor out \(\frac{1}{4}\) from the \(x\) terms: \(y = \frac{1}{4}(x^2 + 6x) + \frac{1}{4}\).

Complete the square inside the parentheses: take half of the coefficient of \(x\) (which is 6), square it, and add and subtract that value inside the parentheses. Half of 6 is 3, and \(3^2 = 9\). So, write \(x^2 + 6x + 9 - 9\) inside the parentheses: \(y = \frac{1}{4}((x + 3)^2 - 9) + \frac{1}{4}\).

Simplify the equation by distributing \(\frac{1}{4}\) and combining constants: \(y = \frac{1}{4}(x + 3)^2 - \frac{9}{4} + \frac{1}{4}\). This gives the standard form of the parabola, from which you can identify the vertex, focus, and directrix.

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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 = k, which helps convert equations into standard form. This 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's equation reveals its vertex and orientation. For a parabola opening vertically, it is written as (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p relates to the distance between the vertex and the focus or directrix.

Vertex, Focus, and Directrix of a Parabola

The vertex is the parabola's turning point, the focus is a fixed point inside the parabola, and the directrix is a line outside it. The parabola is the set of points equidistant from the focus and directrix, and these elements are essential for graphing and understanding its shape.

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 + 6x - 4y + 1 = 0