Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y² = - 8x

검증된 단계별 안내

Identify the form of the given parabola equation. The equation is \(y^2 = -8x\), which matches the standard form of a horizontal parabola: \(y^2 = 4px\).

Compare the given equation \(y^2 = -8x\) with the standard form \(y^2 = 4px\) to find the value of \(p\). Here, \(4p = -8\), so solve for \(p\) by dividing both sides by 4.

Interpret the value of \(p\). Since \(p\) is negative, the parabola opens to the left. The vertex of the parabola is at the origin \((0,0)\) because the equation is not shifted.

Find the focus using the vertex and \(p\). The focus lies at \((p, 0)\), so substitute the value of \(p\) found in step 2 to get the coordinates of the focus.

Find the directrix. The directrix is a vertical line given by \(x = -p\). Use the value of \(p\) to write the equation of the directrix.

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주요 개념

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Standard Form of a Parabola

The standard form of a parabola's equation helps identify its orientation and key features. For parabolas that open left or right, the form is y² = 4px, where p represents the distance from the vertex to the focus. Recognizing this form allows you to extract important information like the focus and directrix.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola used to define it, and the directrix is a line perpendicular to the axis of symmetry. The parabola consists of all points equidistant from the focus and directrix. Knowing how to calculate the focus and directrix from the equation is essential for graphing.

Graphing Parabolas

Graphing a parabola involves plotting the vertex, focus, and directrix, then sketching the curve that is symmetric about the axis of symmetry. Understanding the direction the parabola opens (left, right, up, or down) and the distance p helps in accurately drawing the shape.

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y2 = - 8x