The focus. A fixed point on the interior of the parabola that is used for the formal definition of the curve. The directrix. A fixed, straight line. The parabola is the locus (series) of points in which any given point is of equal distance from the focus and the directrix. (See the diagram above. ) The axis of symmetry. This is a straight line that passes through the turning point (“vertex”) of the parabola and is equidistant from corresponding points on the two arms of the parabola. The vertex. The point where the axis of symmetry crosses the parabola is called the vertex of the parabola. If the parabola opens upward or to the right, the vertex is a minimum point of the curve. If it opens downward or to the left, the vertex is a maximum point.
If the coefficient a in the equation is positive, the parabola opens upward (in a vertically oriented parabola), like the letter “U”, and its vertex is a minimum point. If the a is negative, the parabola opens downward and has a vertex at its maximum point. If you have trouble remembering this, think of it this way: an equation with a positive a value looks like a smile; an equation with a negative a value looks like a frown. [5] X Expert Source Jake AdamsAcademic Tutor & Test Prep Specialist Expert Interview. 20 May 2020. Let’s say you have the following equation: y = 2x2 -1. This parabola will be shaped like a “U” because the a value (2) is positive. If the equation has a squared y term instead of a squared x term, the parabola will be oriented horizontally and open sideways, to the right or left, like a “C” or a backward “C. " For example, the parabola y2 = x + 3 opens to the right, like a “C. "
In the above example (y = 2x² -1), a = 2 and b = 0. Now you can calculate the axis of symmetry by plugging in the numbers: x = -0 / (2)(2) = 0. In this case the axis of symmetry is x = 0 (which is the y-axis of the coordinate plane).
The coordinates of the vertex are sometimes known as (h, k). In this case h is 0, and k is -1. The equation for the parabola may be written in the form y = a(x – h)² + k. In this form the vertex is the point (h, k), and you don’t need to do any math to find the vertex beyond interpreting the graph correctly.
The middle value of x should be the axis of symmetry in the case of a “vertical” parabola. You should include at least two values above and below the middle value for x in the table for the sake of symmetry. In this example, put the value of the axis of symmetry (x = 0) in the middle of the table.
For x = -2, y is calculated as: y = (2) (-2)2 - 1 = 8 - 1 = 7 For x = -1, y is calculated as: y = (2) (-1)2 - 1 = 2 - 1 = 1 For x = 0, y is calculated as: y = (2) (0)2 - 1 = 0 - 1 = -1 For x = 1, y is calculated as: y = (2) (1)2 - 1 = 2 - 1 = 1 For x = 2, y is calculated as: y = (2) (2)2 - 1 = 8 - 1 = 7
The x-axis is horizontal; the y-axis is vertical. The positive numbers on the y-axis are above the point (0, 0), and the negative numbers on the y-axis are below the point (0, 0). The positive numbers on the x-axis are to the right of the point (0, 0), and the negative numbers on the x-axis are to the left of the point (0, 0).
