Question

Find the equation of the parabola defined by the given information. Sketch the parabola. Focus: (-3,0)\(;\) vertex: (0,0)

Short answer

The equation of the parabola is \(y^2 = -12x\). It opens left.

Step-by-step solution

Understand the properties of the parabola

A parabola can be defined by its vertex and focus. Given the vertex (0,0) and focus (-3,0), it is a horizontally opening parabola. This means its standard form is \( (y - k)^2 = 4p(x - h) \). Here, \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus, in this case \(-3\).

Calculate the value of p

To find \(p\), calculate the distance from the vertex to the focus. The vertex is (0,0), and the focus is (-3,0), so \(p = -3 - 0 = -3\).

Write the equation using the standard form

Substitute \(h = 0\), \(k = 0\), and \(p = -3\) into the standard form \((y - k)^2 = 4p(x - h)\). This gives \(y^2 = 4(-3)x\), or \(y^2 = -12x\).

Sketch the parabola

The equation \(y^2 = -12x\) describes a parabola that opens to the left, as indicated by the negative coefficient of \(x\). Since the vertex is at (0,0) and the focus is at (-3,0), the parabola is symmetric along the x-axis, with points stretching out to the left of the y-axis.

Key concepts

Vertex form of parabola

The vertex form of a parabola is an alternative presentation of its equation that highlights the vertex, which is the point where the parabola changes direction. This form is particularly useful for quickly identifying a parabola's vertex from its equation.
The general vertex form of a parabola that opens either up or down is \[(x - h)^2 = 4p(y - k)\]where

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

For parabolas opening left or right, the roles of \(x\) and \(y\) are swapped, leading to the formula:\[(y - k)^2 = 4p(x - h)\].
In the exercise, we used this form with \( (h, k) = (0, 0)\) and \( p = -3 \) to write the equation \( y^2 = -12x \). This equation indicates a parabola with its vertex at the origin and opening sideways to the left.

Focus and Directrix

Understanding the focus and directrix is crucial to the concept of parabolas.
Every parabola is defined by a point called the focus and a line called the directrix. The parabola is the set of all points equidistant from the focus and the directrix.

• The focus is a point that lies inside the parabola and direct the curve of the parabola towards it.
• The directrix is a line that is perpendicular to the line segment connecting the vertex and the focus.

For the exercise, the focus was given as \((-3, 0)\) and the vertex at \((0,0)\), implying that the directrix is a vertical line at \(x = 3\). The relationship between the vertex, focus, and directrix determines the parabola's shape and direction of opening.

Properties of Parabolas

Parabolas have unique geometric properties that make them stand out among conic sections. Some key properties include:

• **Symmetry**: Parabolas are symmetrical around their axis of symmetry, which in the exercise, is the \(y\)-axis.
• **Vertex**: The vertex is the highest or lowest point on the parabola, depending on if it opens upwards or downwards (or neither, for horizontal openings).
• **Opening Direction**: The direction in which a parabola opens is determined by the sign of the \(4p\) term in its equation.
  A negative \(p\) as in \(y^2 = -12x\) means the parabola opens leftwards.
• **Width and Distance**: The \(p\) value not only directs the opening but also affects the "width" or "spread" of the parabola. A larger absolute value of \(p\) means a "wider" parabola.

The parabola in the exercise is defined as \(y^2 = -12x\), emphasizing these properties, like its symmetry about the \(x\)-axis and opening towards the left.