Parabolas

An equation in two variables where one of the variables has a degree of two and the other variable has a degree of one will form a parabola.  The parabola has a single high or low point, called the vertex.  After finding this single point, all other points will either be above or below the vertex depending on the sign of the squared variable term.  If the second degree variable term has a positive coefficient all points will be above the vertex.  If the second degree variable term has a negative coefficient all points will be below the vertex.  For example, , would be a parabola that would rise from the vertex.  On the otherhand, , would be a parabola that would fall from the vertex.  First you must determine the vertex.  The vertex may be found by first finding the x-value of the point which is .  *Note: the a and b are identified by the coefficients of the quadratic function when written in the standard form ...

Once you find the x-value, plug that value back into the equation and calculate the corresponding y-value.  For the parabola , .

Now to find the y-value, substitute the x-value you just calculated back into the equation.  .  Thus, the vertex is (–2, –10).

One way to find other points on the parabola is to find the x-intercepts and y-intercept.  You may be able to factor the polynomial and use the zero factor property to determine x-intercepts.  To determine the y-intercept, let x = 0 and calculate the y-value.  It may be helpful to review solving second quadratic equations.

Another way to find other pairs of points could be to choose two “arbitrary” values for x on one side of the vertex and calculate the corresponding y-values.  Once you have found these points, you can “reflect” those points onto the opposite side of the vertex.  For example, using the same equation where we just found the vertex, suppose we choose x = –1 and x = 0 to be two values for x to the right of the vertex.  Then   and .  Therefore, the points (–1, –9) and (0, –6) are on the graph of the parabola.  The parabola is symmetrical, which means has opposing points on the left side of the vertex with the same y-value.  Since x = –1 is one unit to the right of the vertex, there would be another point one unit to the left of the vertex at x = –3 with the y-value of –9.  Also, since x = 0 is two units to the right of the vertex, there would be another point two units to the left of the vertex at x = –4 with the y-value of –6.  We have now found the points (–3, –9) and (–4, –6).  Graph these five points and connect the points to form the parabola.

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 15-Nov-2007