How to Analyze a Parabola

A parabola is a type of curve that appears in algebraic equations and is defined as the set of all points that are equidistant to a fixed point (the focus) and a fixed line (the directrix). Analyzing a parabola involves understanding its key features, such as the vertex, focus, directrix, axis of symmetry, and intercepts.

Here are the steps to analyze a parabola:

1. Determine the orientation of the parabola: Is it opening up or down? If the coefficient of the squared term is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.

2. Identify the vertex: The vertex is the point where the parabola changes direction. To find the vertex, you can use the formula x = -b/2a to find the x-coordinate, and then substitute it into the equation to find the y-coordinate. Alternatively, you can complete the square to write the equation in vertex form: y = a(x-h)^2 + k, where (h,k) is the vertex.

3. Find the axis of symmetry: The axis of symmetry is a line that divides the parabola into two symmetric parts. It passes through the vertex and is parallel to the directrix. The equation of the axis of symmetry is x = h.

4. Determine the focus and directrix: The focus is a point on the axis of symmetry that is equidistant from the vertex and the directrix. The directrix is a line that is equidistant to all points on the parabola. To find the focus and directrix, you can use the formula 1/4a to determine the distance from the vertex to the focus or directrix, and then use the equation of the axis of symmetry to find their coordinates.

5. Find the x- and y-intercepts: The x-intercepts are the points where the parabola intersects the x-axis, and they occur when y = 0. To find the x-intercepts, set y = 0 in the equation of the parabola and solve for x. The y-intercept is the point where the parabola intersects the y-axis, and it occurs when x = 0. To find the y-intercept, substitute x = 0 into the equation of the parabola and solve for y.

By following these steps, you can analyze a parabola and understand its key features.