When is a parabola upside down

Notice that the x -intercepts of any graph are points on the x -axis and therefore have y -coordinate 0. If the equation factors we can find the points easily, but we may have to use the quadratic formula in some cases. If the solutions are imaginary, that means that the parabola has no x -intercepts is strictly above or below the x -axis and never crosses it. If the solutions are real, but irrational radicals then we need to approximate their values and plot them. In the module Algebra review , we revised the very important technique of completing the square.

Detailed description of diagrams. The graph is shown below. Not all parabolas are congruent to the basic parabola. This transformation of stretching can now be combined with the other transformations discussed above.

Once again, the basic algebraic technique is completing the square. The vertex is the point of the parabola at the axis of symmetry. To find the y -coordinate of the vertex, find the axis of symmetry and substitute that x -value into the original equation. Below is an example of how to calculate the focus and directrix that may provide a better understanding of the mathematical definition of a parabola provided above:.

The focus is a point located on the same line as the axis of symmetry, while the directrix is a line perpendicular to the axis of symmetry. For parabolas, the focus is always on the inside of the parabola, and the directrix never touches the parabola. Because the vertex is the same distance from the focus and directrix, the directrix has a location directly opposite of the focus.

The vertex form of a parabola is:. The a in the vertex form of a parabola corresponds to the a in standard form. If a is positive, the parabola will open upwards.