# Strange tangent for parametric

## Main Question or Discussion Point

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 postive 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

## Answers and Replies

jedishrfu
Mentor
How are you computing the tangents?

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

Mark44
Mentor
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
Just show us your work, not an image of it, especially one that doesn't render.

Mark44
Mentor
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 postive 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
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?

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.

Mark44
Mentor
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.
Right, but they occur for two different values of t.

When t = -1, (x, y) = (2, -4) and the tangent is vertical.
When t = 2 you get (2, -4) again, but this time with a horizontal tangent.

To find the values of t, I solved the equation t3 - 3t = 2, or t3 - 3t - 2 = 0, which in factored form is (t + 1)2(t - 2) = 0.

Thank you so much, I understand now