Problem 75 - A Graphical Approach to College Algebra

Question

Write an equation for each parabola with vertex at the origin. Focus \((0,-2)\)

Step-by-Step Solution

Answer

The equation is \(x^2 = -8y\).

Step 1: Identify the Parabola's Orientation

Since the focus is at \((0, -2)\), the parabola opens downwards.

Step 2: Standard Equation of a Parabola

The standard equation for a vertical parabola (opening up or down) with vertex at the origin is \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus or directrix.

Step 3: Determine the Value of p

Here, the focus is directly 2 units down from the vertex, so \(p = -2\). The negative sign indicates that the parabola opens downwards.

Step 4: Write the Equation

Substitute \(p = -2\) into the equation \(x^2 = 4py\): this gives \[x^2 = 4(-2)y = -8y.\]

Step 5: Final Equation

The equation of the parabola with vertex at the origin and focus at \((0, -2)\) is \(x^2 = -8y\).

Key Concepts

Vertex

The vertex is a crucial point of a parabola, as it represents the tip or the turning point. It is the point where the parabola changes direction. In many cases, specifically in mathematical exercises, we focus on parabolas having their vertex at the origin, which is the point (0, 0).

For a parabola with its vertex at the origin, the vertex is both the minimum (for parabolas opening upwards) or the maximum (for parabolas opening downwards) point. This simplifies calculations and equations since the vertex simplifies to \((0, 0)\).

Remember, the vertex is not just about the location of a point. It also influences the equation of the parabola, centering the formula around \(x\) and \(y\).

Focus

The focus of a parabola is another essential feature that takes the geometrical properties of a parabola from abstract to practical. It is a point located inside the parabola, which is crucial for defining its shape. This point affects how 'wide' or 'narrow' the curve is.

In mathematical terms, the focus is located \(p\) units away from the vertex along the axis of symmetry. For the given exercise, where the focus is at \((0, -2)\), \(p\) is determined as \(-2\). The sign of \(p\) is essential to denote which way the parabola opens (downward in this case because \(p\) is negative).

The focus, paired with the directrix, help define every point on a parabola. For every point on the curve, the distance to the focus equals the perpendicular distance to the directrix.

Standard Equation of a Parabola

Knowing the standard equation of a parabola is vital as it helps solve problems concerning the geometric and algebraic properties of a parabola. For a parabola that opens up or downwards and has its vertex at the origin, the standard form is given by \(x^2 = 4py\).

In this equation, \(4p\) determines both the direction and `width` of the parabola. When \(p\) is positive, the parabola opens upwards, and when \(p\) is negative, it opens downwards. For the parabola in the exercise, substituting \(p = -2\) results in \(x^2 = -8y\).

Recognizing and understanding the standard form helps in sketching parabolas and grasping how altering values affects the curve's shape. It provides insights into the parabola's orientation, focus, and width, facilitating better problem-solving skills.