Question

Identify the focus, directrix, and axis of symmetry of the parabola. Graph the equation. (See Example 2.) $$6 x^2+3 y=0$$

Short answer

The focus of the given parabola is at (0, 1/2), the directrix is the line y=1/2, and the axis of symmetry is x=0. The parabola opens downwards.

Step-by-step solution

Write equation in standard form

The given equation is \(6x^2+3y=0\). We need to rewrite it in the standard form of the parabola equation. Divide the equation by 3 to simplify. This results in \(2x^2+y=0\). Or, rewriting \(y= -2x^2\). Now this equation is in standard form, where \(h=0\), \(k=0\), and \(4a= -2\).

Identify the focus

For the parabola \(y=4ax\), the focus is at \(F=(h, k-a)\). Here, \(a\) is \(-2/4 = -1/2\). So, the focus of the parabola is at \(F=(0, 1/2)\).

Identify the directrix

The directrix of the parabola \(y=4ax\) is the line \(y=k+a\). Here, \(k=0\) and \(a=-1/2\), so the directrix is \(y=1/2\).

Identify the axis of symmetry

The axis of symmetry for a parabola in the form \(y=4ax\) is always the line \(x=h\), and in this case it is \(x=0\).

Sketch the graph

Plot the focus, vertex, and directrix on a graph. The vertex is at the origin. The focus is the point above the vertex, and the directrix is a horizontal line below the vertex. Sketch a rough graph of the parabola.

Key concepts

Focus of a Parabola

The focus of a parabola is a point inside the parabola where all the reflected lines from the curve converge or meet. This unique point is crucial because it helps define the properties and direction of the parabola. For the parabola given, the equation is written as \( y = -2x^2 \). By comparing it with the standard form \( y = 4ax \), we find the parameter \( a \).

• The value \( a \) is calculated as \( -\frac{1}{2} \).
• The focus is at \( F = (h, k-a) = (0, 1/2) \) given our findings from the solution.

To visualize it, imagine a point above the vertex (the turning point of the parabola). This focus is essential for sketching a parabola and helps understand how light or sound might travel inside such a shape.
Remember, no matter where you place a parabola, it will always have one focus point!

Directrix of a Parabola

The directrix is a straight line that is perpendicular to the axis of symmetry of the parabola. It serves as a reference line by which distances are measured to maintain the parabolic shape. This line is crucial, as it helps define how wide or narrow the parabola is.

• For the parabola \(y = -2x^2\), the directrix is found using \( y = k + a \).
• Here, \( k = 0 \) and \( a = -\frac{1}{2} \), resulting in a directrix at \( y = 1/2 \).

Understanding the directrix is essential for plotting a parabola accurately. Picture it as a horizontal line below the vertex of the parabola. This line aids in constructing the curve and ensures that each point on the parabola is equidistant from both the focus and the directrix.
It's fascinating how the directrix works alongside the focus to shape the entire parabola!

Axis of Symmetry

The axis of symmetry for a parabola is an imaginary line that divides the parabola into two mirror-image halves. This line runs vertically (or horizontally, depending on the direction of the parabola) through the vertex and is a guide to ensure the parabola is symmetrical.

• In our equation \(y = -2x^2\), the axis of symmetry can be deduced as \( x = h \).
• With \( h = 0 \), the axis of symmetry for our parabola is \( x = 0 \).

This axis is incredibly useful for graphing a parabola because it helps align the focus and vertex. You can think of it as an invisible line running through the vertex, ensuring every point on one side of the parabola has an exact mirror on the other side.
Recognizing the axis of symmetry is a crucial step in sketching and understanding parabolic curves efficiently.