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Definition

A parabola is a symmetric curve formed by all points equidistant from a point called the focus and a line called the directrix. It is the graph of a quadratic function in the form $y = ax^2 + bx + c$.

5 Must Know Facts For Your Next Test

1. The vertex of a parabola is its highest or lowest point, depending on its orientation.
2. The axis of symmetry is a vertical line that passes through the vertex of the parabola.
3. The focus lies inside the parabola, while the directrix is outside, and they determine its shape.
4. When $a > 0$, the parabola opens upwards; when $a < 0$, it opens downwards.
5. The standard form of a parabolic equation can be converted to vertex form $y = a(x - h)^2 + k$, where $(h,k)$ is the vertex.

Review Questions

• What are the coordinates of the vertex for the parabola given by $y = 2x^2 - 4x + 1$?
• Explain how to find the focus and directrix of a given parabolic equation.
• Discuss how changing the value of 'a' in $y = ax^2 + bx + c$ affects the orientation and width of the parabola.

Related terms

Vertex: The highest or lowest point on a parabola, where it changes direction.

Axis of Symmetry: A vertical line passing through the vertex, dividing the parabola into two symmetrical halves.

$Focus \, \& \, Directrix$: $Focus$: A fixed point inside a parabola. $Directrix$: A fixed line outside a parabola. Together, they define its specific shape.