Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (x + 1)² = - 8(y + 1)

Verified step by step guidance

1. Identify the form of the given equation. The equation is \( (x + 1)^2 = -8(y + 1) \), which matches the standard form of a vertical parabola: \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex.
2. From the equation, determine the vertex \( (h, k) \). Here, \( h = -1 \) and \( k = -1 \), so the vertex is at \( (-1, -1) \).
3. Find the value of \( p \) by comparing \( 4p \) to the coefficient on the right side. Since \( 4p = -8 \), solve for \( p \) to get \( p = -2 \). The negative value indicates the parabola opens downward.
4. Use the vertex and \( p \) to find the focus. The focus lies \( p \) units away from the vertex along the axis of symmetry (vertical line \( x = h \)). So, the focus is at \( (h, k + p) = (-1, -1 - 2) \).
5. Find the directrix, which is a horizontal line \( p \) units in the opposite direction from the vertex. The directrix is \( y = k - p = -1 - (-2) = -1 + 2 \). This line is above the vertex.

Key Concepts

Standard Form of a Parabola

The standard form of a parabola's equation helps identify its orientation and key features. For vertical parabolas, the form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p determines the distance to the focus and directrix. Recognizing this form allows you to extract important information directly from the equation.

Vertex, Focus, and Directrix

The vertex is the parabola's turning point, located at (h, k). The focus is a point inside the parabola at a distance p from the vertex, and the directrix is a line perpendicular to the axis of symmetry, also p units from the vertex but in the opposite direction. These elements define the parabola's shape and position.

Graphing Parabolas

Graphing a parabola involves plotting the vertex, focus, and directrix, then sketching the curve that is equidistant from the focus and directrix. Understanding the orientation (upward, downward, left, or right) based on the equation's form is essential for accurate graphing.