Quick question about parametric curves

renegade05

renegade05

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\(\displaystyle x=t^3-2t+1\)
\(\displaystyle y=t^2-5\)
\(\displaystyle Restriction for both: -3\leq t\leq3\)

Question: Find the exact x intercept(s) of the curve.

I am just drawing a blank for some reason. Usually im pretty good at this stuff.

Do I graph \(\displaystyle x=t^3-2t+1\) and find zeroes ?

I noticed how it said "exact" in the question. How can i solve this cubic anylatically if that is the case?

Need some guidance. Thanks!
 
x-intercepts appear at y = 0. Think about any intercept. What are its coordiantes? There's always a zero in there, somewhere.
 
I hate it when someone tells me something i already know...
Thanks tkhunny. That actually jolted my brain in getting the answer.

\(\displaystyle x=\sqrt{5},-\sqrt{5}\)
 
Hmmm... Better try that again. Is that "x=", as you have written it, or is it "t="? It makes quite a difference.
 
tkhunny said:
Hmmm... Better try that again. Is that "x=", as you have written it, or is it "t="? It makes quite a difference.
Click to expand...

t=
:oops:

..thats why i am "new member" and you sir are ELTIE!
 
I'm still nervous. It's not just a typo. What values for x are the result?
 
There you go. I should have been nervous. That does not answer the question.

x-intercepts look like this (a,0), where 'a' is a Real Number. In this case, it might be okay to write (x(b),0), where 'b' is a Real Number.

It's probably on an set of perpendicular x-,y- coordinate axes. (Cartesian Coordiante System or Cartesian Plane). No 't' in there.
 
tkhunny said:
There you go. I should have been nervous. That does not answer the question.

x-intercepts look like this (a,0), where 'a' is a Real Number. In this case, it might be okay to write (x(b),0), where 'b' is a Real Number.

It's probably on an set of perpendicular x-,y- coordinate axes. (Cartesian Coordiante System or Cartesian Plane). No 't' in there.
Click to expand...

So you are saying the anwer should look like this:

\(\displaystyle (\sqrt{5},0),(-\sqrt{5},0)\)
 
That si the basic form, but you continue to confuse x and t.

Reset!!

You need y = 0. You used this fact to find t = sqrt(5) and t = -sqrt(5)

I don't know if you did this, but you should have checked to see that -3 < -sqrt(5) < 3 and that -3 < -sqrt(5) < 3.

Now, you must use these values of 't' to find the desired values of 'x'.

\(\displaystyle x\left(\sqrt{5}\right)\;=\;\left(\sqrt{5}\right)^{3} - 2\cdot\sqrt{5} + 1\)

This will give you one of the desired x-values.
 
tkhunny said:
That si the basic form, but you continue to confuse x and t.

Reset!!

You need y = 0. You used this fact to find t = sqrt(5) and t = -sqrt(5)

I don't know if you did this, but you should have checked to see that -3 < -sqrt(5) < 3 and that -3 < -sqrt(5) < 3.

Now, you must use these values of 't' to find the desired values of 'x'.

\(\displaystyle x\left(\sqrt{5}\right)\;=\;\left(\sqrt{5}\right)^{3} - 2\cdot\sqrt{5} + 1\)

This will give you one of the desired x-values.
Click to expand...
Wow, I made this way harder than it had to be.

I get it now.

\(\displaystyle x=\;\left(\sqrt{5}\right)^{3} - 2\cdot\sqrt{5} + 1\)
\(\displaystyle x=\;\left(-\sqrt{5}\right)^{3} - 2\cdot(-\sqrt{5}) + 1\)

I cant see anyway of simplifying those, since my teacher wants exact answers. do you?
 
renegade05 said:
tkhunny said:
That si the basic form, but you continue to confuse x and t.

Reset!!

You need y = 0. You used this fact to find t = sqrt(5) and t = -sqrt(5)

I don't know if you did this, but you should have checked to see that -3 < -sqrt(5) < 3 and that -3 < -sqrt(5) < 3.

Now, you must use these values of 't' to find the desired values of 'x'.

\(\displaystyle x\left(\sqrt{5}\right)\;=\;\left(\sqrt{5}\right)^{3} - 2\cdot\sqrt{5} + 1\)

This will give you one of the desired x-values.
Click to expand...
Wow, I made this way harder than it had to be.

I get it now.

\(\displaystyle x=\;\left(\sqrt{5}\right)^{3} - 2\cdot\sqrt{5} + 1 \ = \ \;5\left(\sqrt{5}\right) - 2\cdot\sqrt{5} + 1 \ = \ 3\left(\sqrt{5}\right) + 1\)

\(\displaystyle x=\;\left(-\sqrt{5}\right)^{3} - 2\cdot(-\sqrt{5}) + 1 \ = \;-\left(\sqrt{5}\right)^{3} + 2\cdot(\sqrt{5}) + 1 \ = \ ???\)

I cant see anyway of simplifying those, since my teacher wants exact answers. do you? ? yes I do...
Click to expand...
 
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