Now looking at this vector visually, do you see how we can use the slope of the line of the vector from the initial point to the terminal point to get the direction of the vector? Here is all this visually.
To find the domain and range, make a t-chart: Notice that when we have trig arguments in both equations, we can sometimes use a Pythagorean Trig Identity to eliminate the parameter and we end up with a Conic: Parametric Equations Eliminate the parameter and describe the resulting equation: Eliminate the parameter and describe the resulting equation: Sometimes you may be asked to find a set of parametric equations from a rectangular cartesian formula.
This seems to be a bit tricky, since technically there are an infinite number of these parametric equations for a single rectangular equation. And remember, you can convert what you get back to rectangular to make sure you did it right! Work these the other way from parametric to rectangular to see how they work!
And remember that this is just one way to write the set of parametric equations; there are many!
Here are some examples: Now, the second point: Easier way using vectors: The parametric equations are. The parametric equations are: Try it; it works!
Applications of Parametric Equations Parametric Equations are very useful applications, including Projectile Motion, where objects are traveling on a certain path at a certain time.
It appears that each of the set of parametric equations form a line, but we need to make sure the two lines cross, or have an intersection, to see if the paths of the hiker and the bear intersect.
So that answer to a above is yes, the pathways of the hiker and bear intersect. We can see where the two lines intersect by solving the system of equations: At noon, Julia starts out from Austin and starts driving towards Dallas; she drives at a rate of 50 mph.
Marie starts out in Dallas and starts driving towards Austin; she leaves two hours later Julia leave at 2pmand drives at a rate of 60 mph. The cities are roughly miles apart.
When will Julia and Marie pass each other? How far will they be from Dallas when they pass each other? Projectile Motion Applications Again, parametric equations are very useful for projectile motion applications.
This is called the trajectory, or path of the object. With a quadratic equation, we could also model the height of an object, given a certain distance from where it started.
Now we can model both distance and time of this object using parametric equations to get the trajectory of an object. Problems Solutions Lisa hits a golf ball off the ground with a velocity of 60 ft.
The wind is blowing against the path of the ball at 10 ft. To solve, either use quadratic formula, or put in graphing calculator degree mode: So Lisa hits the golf ball Jade, a pro softball player, hits the ball when it is 3 feet off the ground with an initial velocity of ft.
A straight-line wind is blowing at 14 ft. She hits the ball towards a 40 foot fence that is feet from the plate; if it clears this fence, the ball is a home run. Note also that we had to add the initial height of 3.
So at feet from the home plate, the ball has a height of about There is information on the parametric form of the equation of a line in space here in the Vectors section.Algebra > Lines > Finding the Equation of a Line Given a Point and a Slope.
Page 1 of 2. Let's find the equation of the line that passes through the point (4, -3) Finding the Equation of a Line Given Two Points.
Parallel Lines. Perpendicular Lines.
Help ASAP PLZ FAST Will rate u 1. The graph of a trigonometric function oscillates between y=1 and y= It reaches its maximum at x =pi and its minimum at x=3.
Writing Algebra Equations Finding the Equation of a Line Given Two Points. We have written the equation of a line in slope intercept form and standard form.
We have also written the equation of a line when given slope and a point.
Now we are going to take it one step further and write the equation of a line when we are only given two points that . You can find the straight-line equation using the point-slope form if they just give you a couple points: Find the equation of the line that passes through the points (–2, 4) and (1, 2).
I've already answered this one, but let's look at the process. Great Circles. A great circle is the intersection a plane and a sphere where the plane also passes through the center of the sphere. Lines of longitude and the equator of .
Writing Linear Equations Given Slope and a Point. When you are given a real world problem that must be solved, you could be given numerous aspects of the equation.
If you are given slope and the y-intercept, then you have it made. You have all the information you need, and you can create your graph or write an equation in slope intercept form very easily.