In practicality, this means that wherever \(x\) appears in the equation of the parabola we have, we change it \(y\), and vice versa for \(y\). What happens with the parabolas that open up along the x-axis?īy symmetry, this simply obtained by replacing the roles of \(x\) and \(y\) in the equation of the parabola we already have. Hence, doing a translation, the equation of a general parabola with vertex at the point \((k,h)\), with focus \((k, h a)\) and directrix equal to \(y = h-a\) is Simple! Wherever you have \(x\), you replace it by \(x-k\), and wherever you have \(y\), you replace it by \(x-h\). Well, that is the magic of working with a coordinate system, and all we need to do a translation by the point point \((k,h)\)? But how do you do a translation by \((k,h)\)? Now what happens when instead of having the vertex at the origin, we want to have the vertex at a given point \((k,h)\)? This parabola is the kind of parabola that opens up along the y-axis. We won't into much detail, but we will say the equation of a general parabola with vertex at the origin, with focus \((0, a)\) and directrix equal to \(y = -a\) is We need to identify some crucial elements of the parabola: We have the vertex, the focus and the directrix. There are simple derivations to get the equation of a parabola based on the location of a directrix and the focus, but we will skip the derivation in this introduction.Ĭheck the graph below. In the end, by symmetry, it is easy to realize that those parabolas that "open up" along the y-axis have the same structure as those that "open up" along the x-axis, so it is enough to learn how to handle one type. One thing is important to be mentioned: Using functions and relations, there are the parabolas that "open up" along the \(y\)-axis, and there are the parabolas that "open up"along the \(x\)-axis.
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