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

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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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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.

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Solving Quadratic Equations by Completing the Square

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.

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Parabolas as Conic Sections

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.

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Horizontal Parabolas Example 1

Pratica correlata

Domanda del libro di testo

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

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Graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36

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In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $16 x^{2} - y^{2} + 64 x - 2 y + 67 = 0$