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INTRODUCTION TO PARAMETRIC EQUATIONS**

Example 1:

Find a set of parametric equations for the function

use the parametery = 1 x^{2}.tThis is a quadratic function and its graph is a parabola.

Step 1:

Use the parameter

tStep 2:

Step 3: ExpressLet

x =tyin terms of the parameter

y = 1 (t^{2}) =1t^{2}In conclusion, the set of parametric equations consists of

x =andty =.1t^{2}This can also be expressed as an ordered pair as

where the first expression indicates the(t, 1t^{2})x-coordinates of all points on the graph of the functionand the second expression indicates they = 1 x^{2}y-coordinates.Investigate the direction of movement by graphing the set of parametric equations in desmos.com! For example, you would input

Then replace.(t, 1t^{2})on the left withtand on the right with increasingly larger numbers starting at, say, at10.1Watch the curve develop from right to left.

Example 2:

Let's use the function

again, but this time we will use the parametery = 1 x^{2}.tStep 1:

Use the parameter

tStep 2:

Step 3: ExpressLet

x =tyin terms of the parameter

y =1t^{2}In conclusion, the set of parametric equations consists of

x =andty =.1t^{2}This can also be expressed as an ordered pair as

where the first expression indicates the(t, 1t^{2})x-coordinates of all points on the graph of the functionand the second expression indicates they = 1 x^{2}y-coordinates.Investigate the direction of movement by graphing the set of parametric equations in desmos.com! For example, you would input

Then replace.(t, 1t^{2})on the left withtand on the right with increasingly larger numbers starting at, say, at10.1Watch the curve develop from left to righ.

Example 3:

Find a set of parametric equations for the ellipse

where .4x^{2}+ 9y^{2}= 36This ellipse has its center at the origin.

Step 1:

Change the equation to standard form. Let's first divide both sides by

and reduce to get36This can be written as .

Step 2:

We will now use the

Pythagorean Identitywhich issin^{2}(t) + cos^{2}(t) = 1.[sin(t)]^{2}+ [cos(t)] = 1We are given . But we can also see this by relating the

Pythagorean Identityto the standard form in Step 1.Expressed in terms of

x, this is.x = 3sin(t)This is our first parametric equation.It follows that .

Expressed in terms of

y, this is.y = 2cos(t)This is our second parametric equation.In conclusion, she set of parametric equations consists of

andx = 3sin(t).y = 2cos(t)NOTE: The direction of movement along the curve is clockwise. Convince yourself of this by graphing the set of parametric equations in desmos.com! For example, you would input

,(3 sin(t). Then replace2 cos(t) )with increasingly larger positive numbers and watch the curve develop in clockwise direction.t

Example 4:

Find a set of parametric equations for the ellipse

again, but this time let .4x^{2}+ 9y^{2}= 36Step 1:

Change the equation to standard form. Let's first divide both sides by

and reduce to get36This can be written as .

Step 2:

We will now use the

Pythagorean Identitywhich issin^{2}(t) + cos^{2}(t) = 1.[sin(t)]^{2}+ [cos(t)] = 1We are given . But we can also see this by relating the

Pythagorean Identityto the standard form in Step 1.Expressed in terms of

y, this is.y = 2sin(t)This is our first parametric equation.It follows that .Expressed in terms of

x, this is.x = 3cos(t)This is our second parametric equation.In conclusion, she set of parametric equations consists of

andx = 3cos(t).y = 2sin(t)NOTE: The direction of movement along the curve is counter-clockwise. Convince yourself of this by graphing the set of parametric equations in desmos.com! For example, you would input

,(3 cos(t). Then replace2 sin(t) )with increasingly larger positive numbers and watch the curve develop in counter-clockwise direction.t

Example 5:

Find a set of parametric equations for the circle

where .x^{2}+ y^{2}= 25This circle has its center at the origin.

Step 1:

Change the equation to standard form. Let's first divide both sides by

and reduce to get25This can be written as.

Step 2:

We will now use the

Pythagorean Identitywhich issin^{2}(t) + cos^{2}(t) = 1.[sin(t)]^{2}+ [cos(t)] = 1We are given . But we can also see this by relating the

Pythagorean Identityto the standard form in Step 1.Expressed in terms of

x, this is.x = 5sin(t)This is our first parametric equation.It follows that .

Expressed in terms of

y, this is.y = 5cos(t)This is our second parametric equation.In conclusion, she set of parametric equations consists of

andx = 5sin(t).y = 5cos(t)NOTE: The direction of movement along the curve is clockwise. Convince yourself of this by graphing the set of parametric equations in desmos.com! For example, for the first set, you would input

,(5 sin(t). Then replace5 cos(t) )with increasingly larger positive numbers and watch the curve develop in clockwise direction.t

Example 6:

Find a set of parametric equations for the circle

again, but this time let .x^{2}+ y^{2}= 25Step 1:

Change the equation to standard form. Let's first divide both sides by

and reduce to get25This can be written as.

Step 2:

Pythagorean Identitywhich issin^{2}(t) + cos^{2}(t) = 1.[sin(t)]^{2}+ [cos(t)] = 1Pythagorean Identityto the standard form in Step 1.Expressed in terms of

x, this is.y = 5sin(t)This is our first parametric equation.It follows that .

Expressed in terms of

y, this is.x = 5cos(t)This is our second parametric equation.In conclusion, she set of parametric equations consists of

andx = 5cos(t).y = 5sin(t)NOTE: The direction of movement along the curve is counter-clockwise. Convince yourself of this by graphing the set of parametric equations in desmos.com! For example, for the first set, you would input

,(5 cos(t). Then replace5 sin(t) )with increasingly larger positive numbers and watch the curve develop in counter-clockwise direction.t

Example 7:

Find an equation in

xandygiven the set of parametric equations and . Letconsist of all real numbers.tStep 1:

Solve for

in one of the equations.tLet's use then .

Step 2:

Substitute

for2yin the second equation and simplify.tWe find

or

From our study of conic sections we know that this is the standard equation of a parabola open to the right with vertex at the origin!

Example 8:

Find an equation in

xandygiven the set of parametric equationsandx = 3cos(t)Lety = 4sin(t).be between 0 and .tHere we are presented with parametric equations containing sines and cosines. Therefore, we will use the

Pythagorean Identitywhich issin^{2}(t) + cos^{2}(t) = 1.[sin(t)]^{2}+ [cos(t)] = 1Step 1:

We will isolate the trigonometric ratios in the given parametric equations as follows:

and

Step 2:

Replace

andcos(t)in thesin(t)Pythagorean Identitywhich issin^{2}(t) + cos^{2}(t) = 1as follows:[sin(t)]^{2}+ [cos(t)] = 1or since mathematicians want to see

x-terms first in an equation!From our study of conic sections we know that this is the standard equation of an ellipse since

. Its center is at the origin.a bThis equation can also be written as if we multiply both sides by

.144

Example 9:

Find an equation in x and y given the set of parametric equations

andx = 5cos(t)Lety = 5sin(t).be between 0 and .tHere we are again presented with parametric equations containing sines and cosines. Therefore, we will use the

Pythagorean Identityto change to rectangular form.sin^{2}(t) + cos^{2}(t) = 1Step 1:

We will solve the parametric equation for

andcos(t)as follows:sin(t)and

Step 2:

Replace

andcos(t)in thesin(t)Pythagorean Identityas follows:sin^{2}(t) + cos^{2}(t) = 1From our study of conic sections we know that this is the standard equation of a circle since

. Its center is at the origin.a = bThis equation can also be written as

if we multiply both sides byx^{2}+ y^{2}= 25.25