Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y² - 6x = 0

Verified step by step guidance

1. Rewrite the given equation \(y^2 - 6x = 0\) in the standard form of a parabola. Start by isolating the \(x\) term: \(y^2 = 6x\).
2. Recognize that the equation \(y^2 = 4px\) represents a parabola that opens to the right with vertex at the origin \((0,0)\). Here, \$4p = 6\(, so solve for \)p\( by dividing both sides by 4: \)p = \frac{6}{4} = \frac{3}{2}$.
3. Identify the focus of the parabola. Since the parabola opens to the right, the focus is located at \((p, 0)\), which means the focus is at \((\frac{3}{2}, 0)\).
4. Find the equation of the directrix. For a parabola opening to the right, the directrix is a vertical line given by \(x = -p\). Substitute \(p = \frac{3}{2}\) to get the directrix: \(x = -\frac{3}{2}\).
5. To graph the parabola, plot the vertex at the origin \((0,0)\), the focus at \((\frac{3}{2}, 0)\), and draw the directrix line \(x = -\frac{3}{2}\). Sketch the parabola opening to the right, equidistant from the focus and directrix.

Key Concepts

Standard Form of a Parabola

A parabola can be expressed in standard form to identify its geometric properties easily. For a parabola that opens horizontally, the form is (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p determines the distance to the focus and directrix. Rewriting the given equation into this form helps locate these features.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a line outside it, both equidistant from any point on the curve. For the form (y - k)^2 = 4p(x - h), the focus is at (h + p, k) and the directrix is the vertical line x = h - p. Understanding these definitions is key to finding their coordinates.

Graphing a Parabola

Graphing involves plotting the vertex, focus, and directrix, then sketching the curve that is equidistant from the focus and directrix. Knowing the orientation (horizontal or vertical) and the value of p helps determine the shape and width of the parabola. This visual representation aids in understanding the parabola’s properties.