The discussion centers on the parametric equations x = t³ - 3t and y = t³ - 3t², which yield both vertical and horizontal tangents at the point (2, -4). The vertical tangent occurs when t = -1, while the horizontal tangent occurs at t = 2. This phenomenon is explained by the fact that the graph crosses itself at this point, resulting in two distinct tangents for different values of t. The equation t³ - 3t - 2 = 0, factored as (t + 1)²(t - 2) = 0, confirms these values of t.
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Just show us your work, not an image of it, especially one that doesn't render.Calpalned said:https://mail.google.com/mail/u/0/?ui=2&ik=263fc3780d&view=fimg&th=14b0f97b7ad0fc5a&attid=0.1&disp=inline&safe=1&attbid=ANGjdJ9rFhryHaxukeJKxVFkvC2oxwR748Jb1IOpuC3WKs59gfmKcuR1B_n8TTjb5NhFNTaqw8vUgJYVuaXRo63FYUiD7TCzRXcrnGNGm3MDzoHF8zC1VlC2vVLwC3o&ats=1421902241584&rm=14b0f97b7ad0fc5a&zw&sz=w1273-h532
When t = 2, x = 2 and y = -4, just as you say. And dy/dx = 0 when t = 2, so the tangent is horizontal at (2, -4). Why do you think that the tangent is vertical when t = 2?Calpalned said:For the parametric equations x = t^3 - 3t and y = t^3 - 3t^2 I got that the graph has a vertical tangent when t is = to positive or negative one. And it is horizontal at t = 2. However, this implies that at the point (x,y) = (2, -4) the graph has both a vertical and horizontal tangent. How is this possible? Thanks
Right, but they occur for two different values of t.Calpalned said:My friend from math class actually explained it to me, but thank you for your help. At the point (2, -4), the graph crosses itself, so as a result there are two tangents at that spot.