Find the standard equation of a parabola that has a vertical axis and satisfies the given condit

step 1 We need to find the standard equation of the parabola, given that it has x intercepts at negativ

Find the standard equation of a parabola that has a vertical...

Key Concepts

Parabola with Vertical Axis

A parabola with a vertical axis has its symmetry line parallel to the y-axis, meaning it opens either upward or downward. Its equation typically takes the form y = ax² + bx + c, where the direction it opens is determined by the sign of the coefficient a. This concept is crucial because properties such as vertex, axis of symmetry, and intercepts directly depend on this orientation.

x-intercepts (Roots of a Quadratic)

The x-intercepts of a parabola are the points where the graph crosses the x-axis (i.e., where y equals zero). In quadratic equations, these correspond to the solutions or roots of the equation, and they are key when writing the quadratic in factored form. Their locations reveal important aspects about the parabola's shape and can be used to determine the distance between intercepts.

Vertex

The vertex of a parabola is the point that represents its maximum or minimum value, depending on whether the parabola opens downward or upward. This point is important because it indicates the highest or lowest part of the parabola and lies on the axis of symmetry. Knowing the vertex simplifies the process of writing the equation of the parabola in vertex form, y = a(x - h)² + k, where (h, k) is the vertex.

Standard Equation Forms

The equation of a parabola can be expressed in multiple forms, including the standard form y = ax² + bx + c, the factored form y = a(x - r1)(x - r2) where r1 and r2 are the x-intercepts, and the vertex form y = a(x - h)² + k where (h, k) is the vertex. Choosing the appropriate form based on available information, such as intercepts or vertex, is essential for solving problems involving parabolic equations.

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