Question

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola that opens to the right with directrix \(x=-4\)

Short answer

Given the vertex (0, 0) and the directrix x = -4 for a parabola opening to the right, determine the equation of the parabola. Solution: The equation of the parabola is y^2 = 16x.

Step-by-step solution

Find the distance (p) between the vertex and the directrix

Since the parabola opens to the right, the directrix will be a vertical line on the left side of the parabola. Given the directrix is \(x=-4\) and the vertex is at the origin (0, 0), we can find the distance p between them. p = abs(h - directrix) = abs(0 - (-4)) = 4

Plug the vertex and p into the equation of a parabola that opens to the right

Since the vertex is (h,k) = (0,0) and p = 4, the equation of a parabola that opens to the right will be of the form \((y-k)^2 = 4p(x-h)\) Replacing h, k, and p with the given values: \((y-0)^2 = 4(4)(x-0)\)

Simplify the equation

We can simplify the equation by removing the 0s and multiplying 4 by 4: \(y^2 = 16x\) So, the equation of the given parabola is: \(\boxed{y^2 = 16x}\)

Key concepts

Vertex Form

The vertex form of a parabola's equation is a way of expressing the equation that highlights the vertex. The general expression of the vertex form is given by:

• \((y - k)^2 = 4p(x - h)\) when the parabola opens horizontally (either to the right or left).
• \((x - h)^2 = 4p(y - k)\) when the parabola opens vertically (either upward or downward).

In the vertex form:

• \((h, k)\) represents the vertex of the parabola.
• \(p\) is the distance from the vertex to the focus or to the directrix.

For a parabola with its vertex at the origin \((0,0)\), the vertex form simplifies considerably. If the parabola opens to the right, the equation is \(y^2 = 4px\). If it opens to the left, the equation is \(y^2 = -4px\). Understanding these forms makes it easier to visualize how the parabola is positioned relative to its vertex.

Directrix

The directrix of a parabola is a critical component that helps in defining the shape and orientation of the parabola. It is a fixed line used in conjunction with the focus to give the set of all points that form the parabola. For any point on the parabola, the distance to the focus is equal to the distance to this directrix line.

• For a horizontally opening parabola, the directrix will be a vertical line.
• For a vertically opening parabola, the directrix will be a horizontal line.

In the context of the equation \((y - k)^2 = 4p(x - h)\),

• the directrix is located at \(x = h - p\) for parabolas opening horizontally.
• When the vertex is at the origin \((0,0)\) and the directrix is given by \(x = -4\), it implies that \(p = 4\) units to the left from the vertex.

Understanding the location and properties of the directrix is pivotal in sketching and analyzing parabolas correctly.

Parabolas Opening Direction

The opening direction of a parabola is determined by the equation's structure and the value of the parameter \(p\). Parabolas can open in four possible directions:

• Upward or downward if the equation is in the form \((x - h)^2 = 4p(y - k)\).
• Rightward or leftward if the equation is in the form \((y - k)^2 = 4p(x - h)\).

For horizontal parabolas:

• If \(p > 0\), the parabola opens to the right.
• If \(p < 0\), the parabola opens to the left.

For vertical parabolas:

• If \(p > 0\), the parabola opens upwards.
• If \(p < 0\), the parabola opens downwards.

By analyzing the given equation format before graphing, we can predict whether the parabola stretches out horizontally or vertically and in which direction it will curve. This understanding is crucial when sketching the parabola based on its algebraic equation.