To find the parametric equations of the line passing through the point (-1,2,3) and parallel to the vector <3,0,-1>, we first find the vector equation of the line. Thus, the line has vector equation r=<-1,2,3>+t<3,0,-1>. Hence, the parametric equations of the line are x=-1+3t, y=2, and z=3-t.

The parametric equations of the parabola (y – k)2 = 4a(x – h) are x = h + at2 and y = k + 2at.

Example 1:

1. Find a set of parametric equations for the equation y=x2+5 .
2. Assign any one of the variable equal to t . (say x = t ).
3. Then, the given equation can be rewritten as y=t2+5 .
4. Therefore, a set of parametric equations is x = t and y=t2+5 .

The standard form of an equation of a hyperbola centered at the origin with vertices (±a,0) ( ± a , 0 ) and co-vertices (0±b) ( 0 ± b ) is x2a2−y2b2=1 x 2 a 2 − y 2 b 2 = 1 .

The equations x = a cos ф, y = b sin ф taken together are called the parametric equations of the ellipse x2a2 + y2b2 = 1; where ф is parameter (ф is called the eccentric angle of the point P).

Lesson Summary The parametric equation of a straight line passing through (x1, y1) and making an angle θ with the positive X-axis is given by /frac{x-x_1}{cosθ} = /frac{y-y_1}{sinθ} = r , where r is a parameter, which denotes the distance between (x, y) and (x1, y1).

How to quickly convert to the parametric form of equation of line…

1. Eliminate one variable – in this case −3x−3y+6=0⟹x+y=2.
2. Let one variable be t – in this case y=t.
3. Write t in terms of other variables – in this case it’s t=2−x and t=14−4z2.
4. Write it all in one line: x−2−1=y=z−144−12.
5. Write the parametric form: →r=(2,0,144)+t(−1,1,−12)

From what I can gather, the difference is that a cartesian equation involves both the x and the y coordinates in the same equation (e.g. y = x2) while a parametric equation uses another variable as a ‘go between’ for the two equations (e.g. y = t2 and x = t).

Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y − b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.

A Cartesian equation is an equation in terms of x and y only. To obtain a Cartesian equation from parametric equations we must eliminate t.

A cartesian equation of a curve is simply finding the single equation of this curve in a standard form where xs and ys are the only variables. To find this equation, you need to solve the parametric equations simultaneously: If y = 4t, then divide both sides by 4 to find (1/4)y = t.

The Cartesian coordinates (also called rectangular coordinates) of a point are a pair of numbers (in two-dimensions) or a triplet of numbers (in three-dimensions) that specified signed distances from the coordinate axis.

To convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :

1. x = r × cos( θ )
2. y = r × sin( θ )

Example 1

1. Find the polar coordinates of the point Q(-3, 5) and write down the vector. in both Cartesian and polar forms.
2. The Cartesian form is therefore. = -3i + 5j.
3. Hence the polar form of is cos 2.
4. Therfore the Cartesian form of is -2.
5. Find the vector v of magnitude 2 in the direction of the vector r = 3i – j.

The variable t is called a parameter and the relations between x, y and t are called parametric equations. The set D is called the domain of f and g and it is the set of values t takes. As an example, the graph of any function can be parameterized.

To eliminate the angle parameter, rewrite the parametric equations in terms that can be substituted into a trigonometric identity. To eliminate the angle parameter of the two parametric equations above, rewrite the equations in terms of sin θ and cos θ and use trigonometric identity sin 2 θ + cos 2 θ = 1 .

Eliminate Theta Between The Equations

1. Eliminate theta between the equations: q tan θ + p sec θ = x, p tan θ + q sec θ = y.
2. Solution: Squaring both sides of q tan θ + p sec θ = x we get,
3. Eliminate theta between the equations: cos θ + sin θ = m and sec θ + csc θ = n.
4. OR, If cos θ + sin θ = m and sec θ + csc θ = n, prove that n(m2 – 1) = 2m.
5. Solution:

x cos α+y sin α = p This is known as the normal form of the line.

A curve in the plane is said to be parameterized if the set of coordinates on the curve, (x,y), are represented as functions of a variable t. Namely, x = f(t), y = g(t) t D. where D is a set of real numbers. The variable t is called a parameter and the relations between x, y and t are called parametric equations.

If you require the equation of a tangent to a curve, then you have to differentiate to find the gradient at that point, and then use the formula, (y – y1) = m(x – x1), as before. Example: Find the equation of the normal to the curve y = 3×2 – 2x + 1 at the point (1,2).

How do you decide the size of Tstep? Tstep is the increment that your graph uses to plot each point in creating the graph you see on the screen. As a general rule of thumb, the smaller your step is, the more accurate your graph is going to be.

To graph parametric equations by plotting points, make a table with three columns labeled t , x ( t ) /displaystyle t,x/left(t/right) t,x(t), and y ( t ) /displaystyle y/left(t/right) y(t). Choose values for t in increasing order. Plot the last two columns for x and y.

When there is a third variable, a third parameter on whichx andy depend, parametric equations can be used. To graph parametric equations by plotting points, make a table with three columns labeledt,x(t), t , x ( t ) , andy(t).

Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object.

Parametric equations allow you to actually graph the complete position of an object over time. For example, parametric equations allow you to make a graph that represents the position of a point on a Ferris wheel.

A function with a one-dimensional input and a multidimensional output can be thought of as drawing a curve in space. Such a function is called a parametric function, and its input is called a parameter. Sometimes in multivariable calculus, you need to find a parametric function that draws a particular curve.

Non-parametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term non-parametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance.

The Chi-square test is a non-parametric statistic, also called a distribution free test. Non-parametric tests should be used when any one of the following conditions pertains to the data: The level of measurement of all the variables is nominal or ordinal.

The only non parametric test you are likely to come across in elementary stats is the chi-square test. However, there are several others. For example: the Kruskal Willis test is the non parametric alternative to the One way ANOVA and the Mann Whitney is the non parametric alternative to the two sample t test.

Non-parametric Models are statistical models that do not often conform to a normal distribution, as they rely upon continuous data, rather than discrete values. Non-parametric statistics often deal with ordinal numbers, or data that does not have a value as fixed as a discrete number.

If the mean more accurately represents the center of the distribution of your data, and your sample size is large enough, use a parametric test. If the median more accurately represents the center of the distribution of your data, use a nonparametric test even if you have a large sample size.