Problem 39 - Precalculus Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Opens upward with focus 5 units from the vertex

Step-by-Step Solution

Answer

The equation is \( x^2 = 20y \).

Step 1: Understand the Vertex Form of the Parabola

Since the parabola has its vertex at the origin, the standard form of the equation for this parabola is \( y = ax^2 \). This is the general form for a parabola that opens upward or downward.

Step 2: Use the Focus to Determine 'a'

A parabola that opens upward and has its vertex at the origin is expressed as \( (x-h)^2 = 4p(y-k) \). Since the vertex \((h, k)\) is at the origin \((0,0)\), the equation simplifies to \( x^2 = 4py \). The focus is 5 units above the vertex, which tells us that \( p = 5 \).

Step 3: Substitute the Focus into the Equation

By substituting \( p = 5 \) into the equation \( x^2 = 4py \), we get \( x^2 = 20y \). This equation represents a parabola that opens upward with a focus 5 units from the vertex at origin.

Key Concepts

Vertex Form

The vertex form of a parabola's equation makes it simple to identify the vertex of the parabola, which represents its peak or lowest point. For a parabola opening upward or downward, the vertex form is given by:

• \( y = a(x-h)^2 + k \)

Here, \((h, k)\) is the vertex of the parabola. The parameter \(a\) determines the width and the direction of the opening:

• If \(a > 0\), the parabola opens upward.
• If \(a < 0\), it opens downward.

In our exercise, since the vertex is at the origin (0, 0), the equation simplifies to \( y = ax^2 \). This simplified form aids in easily determining the characteristics and direction of the parabola's opening.

Focus of a Parabola

In a parabola, the focus is a special point located along the axis of symmetry, which dictates the parabola's shape and orientation. The distance from the vertex to the focus is denoted as \(p\), and this distance plays a critical role in the parabola's equation. For a parabola opening upwards or downwards, the relation between focus and equation is expressed as:

• \( (x-h)^2 = 4p(y-k) \)

When the vertex is at the origin \((0, 0)\), it simplifies to:

• \( x^2 = 4py \)

In the given exercise, the focus is 5 units away from the vertex, so \( p = 5 \). By substituting this into the simplified equation, we arrive at \( x^2 = 20y \). This tells us that the parabola opens upwards, feeling the pull towards its focus 5 units away.

Standard Form of a Parabola

The standard form of a parabola is another way to express its equation, particularly useful for recognizing the parabola's general direction and width. As with other forms, it varies whether the parabola opens vertically or horizontally. The vertical standard form is:

• \( y = ax^2 + bx + c \)

With this structure:

• \(a\)
• \(b\)
• \(c\)

These coefficients contribute to the parabola's exact curvature and intersection points with the axes. However, when the vertex is simplified to the origin, solving these coefficients becomes straightforward. For instance, in our problem, because no coefficients other than \(a\) significantly alter the equation, we can focus on the relationship between the focus, the vertex, and the resulting specific equation like \(x^2 = 20y\). This scenario underscores how critical the vertex and focus are in shaping the parabola's form.