Quadratic Equations and Expressions Question
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Grade 11 Quadratic Equations and Expressions CCSS: HSG-GPE.A.2
What is the reason in the seventh question for the focus being (h+p,k) and the directrix being x=h-p, instead of just being arbitrary constants? Would the derivation still have been possible using arbitrary constants?
- Using the fact that the focus and directrix are equal distances, p, from the vertex, (h, k), this results in the more common version of the equation for a parabola. The derivation would still have been possible, but the resulting form would not have looked familiar.
- Knowing that the common final form of the equation has the letter p in it, this is included in the focus and directrix, adding in one and subtracting in the other so that they cancel out. The derivation cannot be done without this alteration.
- Since the axis of symmetry of the parabola is perpendicular to the directrix, we use this fact to come up with the p value used in these definitions of the focus and directrix. The derivation would have been identical without this, but is usually included as an interesting mathematical fact.
- There is no reason for this, these are completely arbitrary.