Finding the Standard Equation of a Parabola In Exercises...

Finding the Standard Equation of a Parabola In Exercises $17-24$ , find the standard form of the equation of the parabola with the given characteristics.
Axis is parallel to $y$ -axis; graph passes through $(0,3),(3,4),$ and $(4,11) .$

Key Concepts

Parabola

A parabola is a type of conic section that can be defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This fundamental shape appears in various areas of mathematics and physics, and its geometric properties are essential for studying curves and trajectories.

Standard Equation of a Parabola with Vertical Axis

When the axis of a parabola is parallel to the y-axis, its equation can be written in the standard form y = ax² + bx + c. This form makes it straightforward to analyze various attributes of the parabola, such as its vertex, axis of symmetry, and direction of opening, by directly interpreting the coefficients.

System of Equations

Determining the specific coefficients in a polynomial equation often involves setting up a system of equations. In the context of a parabola passing through given points, substituting the coordinates of these points into the standard equation yields a system of linear equations. Solving this system allows us to identify the unique values for a, b, and c that define the parabola.

Point Substitution in Coordinate Geometry

Using the coordinates of known points to substitute into the equation of a curve is a crucial technique in coordinate geometry. This method not only verifies whether the points lie on the curve but also provides a means to determine unknown parameters within the equation, ensuring the curve accurately fits through the given points.

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