Quadratic and Cubic Equations

Ronald Aguilar

Considering graphs in the xc planes using

Lets start by using different coefficients for a

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Now, lets use different coefficients for b

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As you can see, no matter what the vertices are the parabolas all go through the origin (0,0) and intersect one another. To get a general idea of why this is, I looked at each equation and of course noticed that every equation has . If we were to solve for y, we simply substract y from both sides of the equation. I concluded thats why all the parabolas showed negative characteristics for parabolas. All the parabolas have a maximum point. Lets graph the equation to see what happens to the intersections of the graph.

The parabola goes through every vertices. Lets use the original quadratic equation . In order to get the vertices of this equation and any other equation we must complete the square. In all this is expressed into vertex form:

If we go back through and complete the square for each equation we get the vertices to be

Where the axis of symmetry is x =. If we were to "fold" the parabola in half this where we "fold" the parabola.

In summary, notice that y-coordinate is the square of the x-coordinate. That is why the vertices touch the parabola .

, (h,k) is the vertex.