Question

Determine the equation in standard form of the parabola that satisfies the given conditions. Opens to the right; vertex at (-3,2)$;$ passes through the point (5,10)

   Determine the equation in standard form of the parabola that satisfies the given conditions.
Opens to the right; vertex at (-3,2)$;$ passes through the point (5,10)

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 9, Problem 60

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Solved by Expert Joseph Lentino

08/26/2020

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So, we have $(y-2)^2=4a(x+3)$. Show more…

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Determine the equation in standard form of the parabola that satisfies the given conditions. Opens to the right; vertex at (-3,2)$;$ passes through the point (5,10)

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Key Concepts

Parabola

A parabola is a conic section defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). This curve exhibits symmetry about an axis that passes through its vertex, making it a fundamental object in algebra and geometry.

Vertex Form of a Parabola

The vertex form of a parabola's equation provides an immediate understanding of its geometric transformation, particularly the location of the vertex. For a horizontally oriented parabola, the equation is written as (y - k)² = 4p(x - h), where (h, k) is the vertex. This form is especially useful when given the vertex and additional information that helps determine the parameter p.

Opening Direction

The opening direction of a parabola is determined by the parameter p in its vertex form. If the parabola is horizontal, a positive p indicates that the parabola opens to the right, while a negative p indicates it opens to the left. For vertical parabolas, a positive p means it opens upward and a negative p means it opens downward.

Using a Point to Determine p

When a parabola passes through a point other than the vertex, substituting the coordinates of that point into the vertex form equation allows one to solve for the parameter p. This technique is central to fully defining the parabola’s equation when additional conditions are provided.

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