Parametric Equations

Section 9.5 Parametric Equations Notes

You will be completing the notes for 9.5 together with this Quizizz. Please have the notes in front of you and write as you go.

Parametric Equations

Parametric equations introduce a third variable (the parameter) into the 2-dimensional coordinate plane. 

This third variable is called the parameter.  It is often time.

​

We can now write x as a function of time and y as a function of time to obtain the parametric equation.

Read the paragraphs on your notesheet and click next when you're done. ​

AMA 9.5 Notes

Page 1

Follow the instructions on the bottom of page 1.

1. Mode: PAR

2. y=: enter equations

3. Window: enter settings

4. Graph​

​

Example 1:

Graphing Parametric Equations with your TI-84​

​

Open Ended

Change the T-step to 0.5, to 1.0, -0.5, . . . . What do you notice?

Open Ended

What if the parametric t restriction changes?  What do you notice?

Fill in the table on page 2 for

​-1 ≤ t ≤ 2.

​Evaluate x and y by plugging in values of t:

​When t=-1:

x(-1) = -2(-1) = 2

y(-1) = 4(-1)²​-4 = 0

​

So the first point on the curve is (2,0).​

Graphing by Hand: t-step of 1

​

Keep evaluating functions x(t) and y(t) until you complete the table.

Plot the points on the graph.

​

This is a directed ​graph: it goes from (2,0) towards the left.

​

Use arrows on the graph to denote direction.​

​When you're done with the graph and table, go to the next slide.

Check your table and graph

Multiple Choice

What is the last point on the curve?

• (-4,12)

• (2,0)

• (2,-4,12)

• t=2

Fill in the Blanks

Indicate the domain of the function:

Domain is still the x-values! Use interval notation.

1. ​Solve x = -2t for x

2. Substitute result for t in other equation

3. Simplify and graph the function (you should have an equation for y in terms of x only (t was eliminated!)

Goal: Get Rid of t

Eliminating Parameters 

Fill in the Blanks

When you solve $x=-2t$  for t, what is the result?

Multiple Choice

What is the resulting equation for y in terms of x?

• $$y=x^2-4$$  

• $$y=-x^2-4$$  

• $$y=t^2-4$$  

• $$t=\sqrt[]{\frac{y+4}{4}}$$  

Parameter Eliminated! Check your work:​

Subject | Subject

Some text here about the topic of discussion

Multiple Choice

What is the domain of your new function after eliminating t?

• $$\left[-4,2\right]$$  

• $$\left(-\infty,\infty\right)$$  

• $$\left[-4,\infty\right)$$  

• None of these

What are the differences?

Subject | Subject

Some text here about the topic of discussion

Practice Time!

Turn to the final page and graph the parametric equation using your calculator and by hand! Follow the steps and answer the questions as you go. ​

Follow the instructions on the bottom of page 1.

1. Mode: PAR and RADIAN

2. y=: enter equations

3. Window: enter settings

4. Graph​

​

Example 2:

Graphing Parametric Equations with your TI-84​

​

Fill in the table for ​0 ≤ t ≤ 2π.

​Evaluate x and y by plugging in values of t:

​When t=-1:

x(0) = 3sin(0) = 0

y(0) = 2cos(0) = 2

​

Graphing by Hand: t-step of 1

​

Keep evaluating functions x(t) and y(t) until you complete the table.

Plot the points on the graph.

​

This is a directed ​graph:

Use arrows on the graph to denote direction.​

​When you're done with the graph and table, go to the next slide.

Multiple Choice

Which of the following are the first and last points on the curve?

• (0,2) and (0,-2)

• (3,0) and (3,0)

• (3,0) and (-3,0)

• (0,2) and (0,2)

​Replace this with your body text.

​Duplicate this text as many times as you would like.

Have a nice day. Happy teaching!

​​Subheader

​Replace this with your body text.

​Duplicate this text as many times as you would like.

Have a nice day. Happy teaching!

​​Subheader

Open Ended

Using your calculator, switch start and end of the parameters. You also must make the T_step negative. Aka to:

T_min = 2 $$\pi$$   , T_max = 0, T_step= -0.1308

What happens to the graph?

Options to change direction:

Some text here about the topic of discussion

Change the Equations

Since you swap the first point and the last point, the parametric equations travels in the reverse direction.

​

Switch Start and End of t interval

Multiple Choice

Last Question on the Notes: Eliminating the Parameters

Normally, we solve the equations to isolate t and substitute.

This time, solve each equation to isolate the trig function.

Which is the resulting system?

• $$\frac{x}{3}=t$$   $$y=\frac{2x}{3}$$  

• $$3x=\sin\left(t\right)$$   $$2x=\cos\left(t\right)$$  

• $$\frac{x}{3}=\sin\left(t\right)$$   $$\frac{y}{2}=\cos\left(t\right)$$  

• None of these

We now use everyone's favorite:

​The Pythagorean Identity!!!!

​

​

​

We can eliminate t from the system.

​

The resulting equation is the equation of an ellipse, which matches our graph!​​

Try this out then go to the next slide...

​The final equation is given below.

​

Eliminating with sin/cos

Multiple Choice

Converting parametric equations to a rectangular equation is called _________________ the parameter.

• eliminating
• finding
• converting
• jumping

Multiple Choice

Find the point based on the parametric equations. t = 3

x = 1 - 2t

y =4t + 1

• (-5, 13)

• (13, -5)

• (5, 13)

• (13, 5)

Multiple Choice

Sketch the curve of x(t) = 2t + 1 and y(t) = t² + 2.

• A
• B
• C
• D

Multiple Choice

Eliminate the parameter.

x= 2+4t and y=-1+6t

• y=(3/2)x - 4

• t=(x-2)/4

• y=x - 4

• y= (2/3)x + 4

Multiple Choice

Write the rectangular equation for the following parametric equations.

x = 4cosθ

y = 3sinθ^

• $$\frac{x^2}{9}+\frac{y^2}{16}=1$$

• $$\frac{y^2}{9}+\frac{x^2}{16}=1$$

• $$\cos^2\theta+\sin^2\theta=1$$

• $$\frac{x^2}{9}-\frac{y^2}{16}=1$$