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

  • MHB
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In summary, based on the given parametric equations, the graphs for 21, 23, and 25 can be matched to their respective equations by considering the effect of the parameter t on the shapes. For 21, the circle in the $xz$ plane is affected by an increasing radius, resulting in a circle with an increasing radius. For 23, the circle in the $xy$ plane is affected by an increasing height, resulting in a helix that climbs exponentially. And for 25, the circle in the $xy$ plane has a fixed radius and an exponentially increasing height, resulting in a helix that climbs exponentially.
  • #1
ineedhelpnow
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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!

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  • #2
Let's look at the first one, 21.) What will its "shadow" in the $xz$ plane look like?
 
  • #3
i don't know. i don't understand the question.
 
  • #4
ineedhelpnow said:
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?
 
  • #5

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  • #6
No, what if you had:

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

And then how would the factor of $t$ affect this?
 
  • #7
im getting confused. x=cos t and z=sin t would make a circle.
 
  • #8
ineedhelpnow said:
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?
 
  • #9
i don't know. it affects the amplitude?
 
  • #10
ineedhelpnow said:
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?
 
  • #11
a circle with an increasing radius :confused:
 
  • #12
ineedhelpnow said:
a circle with an increasing radius :confused:

Think of attempting to draw a circle, but the radius keeps growing as you wind around the center...what will you draw instead?
 
  • #13
thanks. i think i figured them out though
 
  • #14
ineedhelpnow said:
thanks. i think i figured them out though

Well, alrighty then. (Smoking)
 
  • #15
i got the first and second but I am still stuck on the last one (25)
 
  • #16
It is similar to the first one. If you ignore $z$, what would you have in the $xy$-plane?
 
  • #18
the z was meant to be an x but its still a circle
 
  • #19
ineedhelpnow said:
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?
 
  • #20
as t increases so does z so I am guessing a circle on the xy plane that increases across z so IV
 
  • #21
Yes, since $z$ increases exponentially, we then expect the helix to climb exponentially. So IV does seem to fit the bill. :D
 

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

What are parametric equations?

Parametric equations are a set of equations that express a set of variables as functions of one or more independent variables. They are used to describe the relationship between multiple quantities and are commonly used in fields such as physics, engineering, and mathematics.

How do I match parametric equations with graphs?

To match parametric equations with graphs, you need to understand the relationship between the independent variable and the dependent variables. The independent variable is usually denoted by t and represents time, while the dependent variables are represented by x and y. Plotting the values of x and y at different values of t will give you points on the graph, and connecting these points will give you the graph of the parametric equations.

What is the significance of the numbers 21, 23, and 25 in this context?

The numbers 21, 23, and 25 are most likely the values of the independent variable t at which the parametric equations are evaluated. These numbers represent specific points on the graph and can help you visualize the shape and behavior of the graph.

How can I use parametric equations in real-world applications?

Parametric equations are used in many real-world applications, such as designing roller coasters, predicting the trajectory of a projectile, and modeling the motion of planets. They are also used in computer graphics to create complex and realistic animations. In general, parametric equations are useful in any situation where multiple quantities are related and can be described as functions of an independent variable.

Are there any limitations to using parametric equations?

While parametric equations are useful in many situations, they do have some limitations. One limitation is that they can only describe linear relationships between variables. Non-linear relationships, such as exponential or trigonometric functions, cannot be represented by parametric equations. Additionally, the values of the independent variable t must be within a certain range for the equations to be valid, so they may not accurately represent all possible values of the dependent variables.

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