This investigation is focused on how to solve quadratic functions by putting them into a perfect square.  The basic form of a quadratic function is y = ax² + bx + c. When drawn on a graph these functions create a parabola that opens down or opens up. The ‘a’ in the function refers to the leading coefficient; the value of this number decides wheather the parabola will be negative or positive. Positive parabolas open up whereas negative parabolas open down.

Here are a few graphs that illustrate positive parabolas that shift vertically due to a different value of ‘c’. The scale of all the graphs is 1 on both the y-axis and x-axis.

A        B        C

y = x²                 y = x² + 3        y = x² -2

The “vertex” is the co-ordinate where the parabola turns; this is also referred to as the “turning point”. In these three graphs we can see a clear transition of the parabola on the vertical y-axis. If the value being added to the ax² is positive the parabola shifts up, above the origin of the x-axis. Thusly if the value being added to the ax² is negative the parabola shifts down under the origin of the x-axis. The only exception would be where nothing is being added to the x² (A) in which case the parabola would have its vertex’s turning point on the origin.

The first graph (A) shows us that y = x² gives us a parabola with the vertex on the origin, the positive leading coefficient is shown by the parabola opening up meaning the parabola is positive. This graph is showing neither dilations, translations nor reflections, because no other coefficients are available.

Nevertheless graph B (y = x² + 3) and graph C (y = x² - 2) show clear indications of vertical translations. B has three units upward and C two units downward because a second coefficient appeared in both equations ("+3", "-2").
Generally speaking if a number (second coefficient "+3" or "-2") is added to the parent function (y = x²), the result is a translation either up or down the y-axis.

Therefore a value added to x² translates the graph up or down on the y-axis.

A different form of the quadratic function is to put it into a perfect square. A perfect square is an equation where the square root to the whole of one side is taken to get the answer on the other side.

eg:

y = (x + b)²

This is a perfect square where if the square root of ‘x + b’ was taken then the answer to ‘y’ would be established.

Here are some graphed examples of perfect squares:

D        E        F

y = ...