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Engineering Calculus Notes 144

Engineering Calculus Notes 144 - equation y = − x 2 whose...

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132 CHAPTER 2. CURVES x 2 4 + y 2 1 = 1 ( x +2) 2 4 + ( y 2) 2 1 = 1 Figure 2.10: Displacing an ellipse We can also reflect a locus about a coordinate axis. Since our model ellipses and hyperbolas are symmetric about these axes, this has no effect on the curve. However, while the model parabola given by Equation ( 2.10 ) is symmetric about the y -axis, it opens up ; we can reverse this, making it open down , by replacing y with y , or equivalently replacing the positive coefficient p with its negative. For example, when p = 1 this leads to the
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Unformatted text preview: equation y = − x 2 whose locus opens down : it is the refection o³ our original parabola y = x 2 about the x-axis (Figure 2.11 ). y = x 2 y = − x 2 Figure 2.11: Refecting a parabola about the x-axis Finally, we can interchange the two variables; this e±ects a refection about the diagonal line y = x . We have seen the e±ect o³ this on an ellipse and hyperbola. For a parabola, the interchange x ↔ y takes the parabola...
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