8.6 Parametric equations  (Page 2/6)

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However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.

Parametric equations

Suppose $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is a number on an interval, $\text{\hspace{0.17em}}I.\text{\hspace{0.17em}}$ The set of ordered pairs, $\text{\hspace{0.17em}}\left(x\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\left(t\right)\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}x=f\left(t\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=g\left(t\right),$ forms a plane curve based on the parameter $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ The equations $\text{\hspace{0.17em}}x=f\left(t\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=g\left(t\right)\text{\hspace{0.17em}}$ are the parametric equations.

Parameterizing a curve

Parameterize the curve $\text{\hspace{0.17em}}y={x}^{2}-1\text{\hspace{0.17em}}$ letting $\text{\hspace{0.17em}}x\left(t\right)=t.\text{\hspace{0.17em}}$ Graph both equations.

If $\text{\hspace{0.17em}}x\left(t\right)=t,\text{\hspace{0.17em}}$ then to find $\text{\hspace{0.17em}}y\left(t\right)\text{\hspace{0.17em}}$ we replace the variable $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with the expression given in $\text{\hspace{0.17em}}x\left(t\right).\text{\hspace{0.17em}}$ In other words, $\text{\hspace{0.17em}}y\left(t\right)={t}^{2}-1.$ Make a table of values similar to [link] , and sketch the graph.

$t$ $x\left(t\right)$ $y\left(t\right)$
$-4$ $-4$ $y\left(-4\right)={\left(-4\right)}^{2}-1=15$
$-3$ $-3$ $y\left(-3\right)={\left(-3\right)}^{2}-1=8$
$-2$ $-2$ $y\left(-2\right)={\left(-2\right)}^{2}-1=3$
$-1$ $-1$ $y\left(-1\right)={\left(-1\right)}^{2}-1=0$
$0$ $0$ $y\left(0\right)={\left(0\right)}^{2}-1=-1$
$1$ $1$ $y\left(1\right)={\left(1\right)}^{2}-1=0$
$2$ $2$ $y\left(2\right)={\left(2\right)}^{2}-1=3$
$3$ $3$ $y\left(3\right)={\left(3\right)}^{2}-1=8$
$4$ $4$ $y\left(4\right)={\left(4\right)}^{2}-1=15$

See the graphs in [link] . It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ increases. (a) Parametric   y ( t ) = t 2 − 1   (b) Rectangular   y = x 2 − 1

Construct a table of values and plot the parametric equations: $\text{\hspace{0.17em}}x\left(t\right)=t-3,\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\left(t\right)=2t+4;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-1\le t\le 2.$

 $t$ $x\left(t\right)$ $y\left(t\right)$ $-1$ $-4$ $2$ $0$ $-3$ $4$ $1$ $-2$ $6$ $2$ $-1$ $8$ Finding a pair of parametric equations

Find a pair of parametric equations that models the graph of $\text{\hspace{0.17em}}y=1-{x}^{2},\text{\hspace{0.17em}}$ using the parameter $\text{\hspace{0.17em}}x\left(t\right)=t.\text{\hspace{0.17em}}$ Plot some points and sketch the graph.

If $\text{\hspace{0.17em}}x\left(t\right)=t\text{\hspace{0.17em}}$ and we substitute $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ into the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ equation, then $\text{\hspace{0.17em}}y\left(t\right)=1-{t}^{2}.\text{\hspace{0.17em}}$ Our pair of parametric equations is

$\begin{array}{l}x\left(t\right)=t\\ y\left(t\right)=1-{t}^{2}\end{array}$

To graph the equations, first we construct a table of values like that in [link] . We can choose values around $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ from $\text{\hspace{0.17em}}t=-3\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}t=3.\text{\hspace{0.17em}}$ The values in the $\text{\hspace{0.17em}}x\left(t\right)\text{\hspace{0.17em}}$ column will be the same as those in the $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ column because $\text{\hspace{0.17em}}x\left(t\right)=t.\text{\hspace{0.17em}}$ Calculate values for the column $\text{\hspace{0.17em}}y\left(t\right).\text{\hspace{0.17em}}$

$t$ $x\left(t\right)=t$ $y\left(t\right)=1-{t}^{2}$
$-3$ $-3$ $y\left(-3\right)=1-{\left(-3\right)}^{2}=-8$
$-2$ $-2$ $y\left(-2\right)=1-{\left(-2\right)}^{2}=-3$
$-1$ $-1$ $y\left(-1\right)=1-{\left(-1\right)}^{2}=0$
$0$ $0$ $y\left(0\right)=1-0=1$
$1$ $1$ $y\left(1\right)=1-{\left(1\right)}^{2}=0$
$2$ $2$ $y\left(2\right)=1-{\left(2\right)}^{2}=-3$
$3$ $3$ $y\left(3\right)=1-{\left(3\right)}^{2}=-8$

The graph of $\text{\hspace{0.17em}}y=1-{t}^{2}\text{\hspace{0.17em}}$ is a parabola facing downward, as shown in [link] . We have mapped the curve over the interval $\text{\hspace{0.17em}}\left[-3,\text{\hspace{0.17em}}3\right],$ shown as a solid line with arrows indicating the orientation of the curve according to $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ Orientation refers to the path traced along the curve in terms of increasing values of $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ As this parabola is symmetric with respect to the line $\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}$ the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ are reflected across the y -axis.

Parameterize the curve given by $\text{\hspace{0.17em}}x={y}^{3}-2y.$

$\begin{array}{l}x\left(t\right)={t}^{3}-2t\\ y\left(t\right)=t\end{array}$

Finding parametric equations that model given criteria

An object travels at a steady rate along a straight path $\text{\hspace{0.17em}}\left(-5,\text{\hspace{0.17em}}3\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\left(3,\text{\hspace{0.17em}}-1\right)\text{\hspace{0.17em}}$ in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.

The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The x -value of the object starts at $\text{\hspace{0.17em}}-5\text{\hspace{0.17em}}$ meters and goes to 3 meters. This means the distance x has changed by 8 meters in 4 seconds, which is a rate of or $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\text{m}/\text{s}.\text{\hspace{0.17em}}$ We can write the x -coordinate as a linear function with respect to time as $\text{\hspace{0.17em}}x\left(t\right)=2t-5.\text{\hspace{0.17em}}$ In the linear function template $\text{\hspace{0.17em}}y=mx+b,2t=mx\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}-5=b.$

Similarly, the y -value of the object starts at 3 and goes to $\text{\hspace{0.17em}}-1,\text{\hspace{0.17em}}$ which is a change in the distance y of −4 meters in 4 seconds, which is a rate of or $\text{\hspace{0.17em}}-1\text{m}/\text{s}.\text{\hspace{0.17em}}$ We can also write the y -coordinate as the linear function $\text{\hspace{0.17em}}y\left(t\right)=-t+3.\text{\hspace{0.17em}}$ Together, these are the parametric equations for the position of the object, where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ are expressed in meters and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ represents time:

$\begin{array}{l}x\left(t\right)=2t-5\hfill \\ y\left(t\right)=-t+3\hfill \end{array}$

Using these equations, we can build a table of values for $\text{\hspace{0.17em}}t,x,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y$ (see [link] ). In this example, we limited values of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ to non-negative numbers. In general, any value of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ can be used.

$t$ $x\left(t\right)=2t-5$ $y\left(t\right)=-t+3$
$0$ $x=2\left(0\right)-5=-5$ $y=-\left(0\right)+3=3$
$1$ $x=2\left(1\right)-5=-3$ $y=-\left(1\right)+3=2$
$2$ $x=2\left(2\right)-5=-1$ $y=-\left(2\right)+3=1$
$3$ $x=2\left(3\right)-5=1$ $y=-\left(3\right)+3=0$
$4$ $x=2\left(4\right)-5=3$ $y=-\left(4\right)+3=-1$

From this table, we can create three graphs, as shown in [link] . (a) A graph of   x   vs.   t ,   representing the horizontal position over time. (b) A graph of y vs.   t ,   representing the vertical position over time. (c) A graph of   y   vs.   x ,   representing the position of the object in the plane at time   t .

find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
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Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what By By  By Anonymous User        