MHB Match Parametric Equations with Graphs - Explanation Help for 21, 23, and 25

[ineedhelpnow]

match the parametric equations with the graphs. i need help with 21 23 and 25. i have the answers but i need explanations for why that specific graph matches with the set of equations. I've been stuck on these for a long time. please help!

View attachment 3089View attachment 3090

 

[MarkFL]

Let's look at the first one, 21.) What will its "shadow" in the $xz$ plane look like?

 

[ineedhelpnow]

i don't know. i don't understand the question.

 

[MarkFL]

i don't know. i don't understand the question.

Okay, ignore $y$ for the moment, what would the curve then defined by the parametric equations for $x$ and $z$ look like in the $xz$ plane?

 

[ineedhelpnow]

View attachment 3091

 

[MarkFL]

No, what if you had:

$$\begin{cases}x=\cos(t) \\[3pt] z=\sin(t) \\ \end{cases}$$

And then how would the factor of $t$ affect this?

 

[ineedhelpnow]

im getting confused. x=cos t and z=sin t would make a circle.

 

[MarkFL]

im getting confused. x=cos t and z=sin t would make a circle.

Yes, good! :D

So, then what do you think the $t$ in front of the trig functions would do to the circle?

 

[ineedhelpnow]

i don't know. it affects the amplitude?

 

[MarkFL]

i don't know. it affects the amplitude?

Yes, that's one way to look at it, I think of it affecting the radius, it is in effect a circle with an increasing radius, which begins at zero, and increases linearly. So, what would we call a circle whose radius is increasing?

 

[ineedhelpnow]

a circle with an increasing radius

 

[MarkFL]

a circle with an increasing radius

Think of attempting to draw a circle, but the radius keeps growing as you wind around the center...what will you draw instead?

 

[ineedhelpnow]

thanks. i think i figured them out though

 

[MarkFL]

thanks. i think i figured them out though

Well, alrighty then. (Smoking)

 

[ineedhelpnow]

i got the first and second but I am still stuck on the last one (25)

 

[MarkFL]

It is similar to the first one. If you ignore $z$, what would you have in the $xy$-plane?

 

[ineedhelpnow]

its a circle. i wanted to upload a pic but the uploader is being dumb graph z'=' cos 8t and y'=' sin 8t - Wolfram|Alpha

 

[ineedhelpnow]

the z was meant to be an x but its still a circle

 

[MarkFL]

its a circle. i wanted to upload a pic but the uploader is being dumb graph z'=' cos 8t and y'=' sin 8t - Wolfram|Alpha

Yes, it's indeed a circle with a fixed radius of one unit, so we know we will have a helical curve of some sort. What does the equation for $z$ tell us about how the helix will climb?

 

[ineedhelpnow]

as t increases so does z so I am guessing a circle on the xy plane that increases across z so IV

 

[MarkFL]

Yes, since $z$ increases exponentially, we then expect the helix to climb exponentially. So IV does seem to fit the bill. :D