Enter your Sign on user name and password.

Forgot password?
• Follow us on:
Start learning today, and be successful in your academic & professional career. Start Today!
Loading video...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Calculus BC

• ## Related Books

Lecture Comments (11)
 0 answersPost by Mohamed E Sowaileh on July 10 at 08:44:36 AMHello Dr. John Zhu, I hope you are very well.I am a student who is extremely weak in math. In order to be very strong in math, specially for engineering field, could you provide me with sequential order of mathematical topics and textbooks. With what should I begin so that I can master big topics like calculus, statistics, probability ... etc.Your guidance is precious to me.Thank you so much. 0 answersPost by Firebird wang on November 2, 2016Are there any way to watch the two video which called AP Statistics Practice Test 2013 an AP Statistics Practice Test 2014? 3 answersLast reply by: Firebird wangWed Oct 12, 2016 10:47 PMPost by Jimmy Jones on September 8, 2015Hello Professor,I am contacting to know if I can go straight to learning this course instead of learning Calculus AB. Is that made possible through these lessonsThanks 0 answersPost by Brady Dill on July 13, 2014Great video! But I was thrown by your use of 'comprise', which means something entirely different. https://www.google.com/search?q=comprise+definition&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a&channel=sb Otherwise, well-explained! I understand now. 0 answersPost by John Zhu on August 12, 2013Yes folks, Ln(0) is never defined! That is a drawing error that SHOULD include a vertical asymptote as Sumant has suggested. Thanks for the feedback and notes guys. Cheers! =) 0 answersPost by Him Tam on July 30, 2013Yeah, the graph shouldn't intersect the y-axis 0 answersPost by Sumant Nigam on March 8, 2013I thought the video was great, but for the last problem: f(x)=ln(x)-1 the graph should not intersect the y-axis, right? There should be a vertical asymptote there. 0 answersPost by Maimouna Louche on June 15, 2012Yey I am first! I loved it all!

### Parametric Curves

• Parametric equation: takes in 2 dependent variables
• Graphing: treat each dependent input variable separately
• Converting to Cartesian equation
• Solve for t in terms of x
• Substitute solved tterm into yterm

### Parametric Curves

Graph f(t) = (3t + 6, cos(t)), for − 2 ≤ t ≤ 2
• Make a table
•  t
 − 2
 − 1
 0
 1
 2
 x(t)
 − 2
 1
 4
 7
 10
 y(t)
 − 0.42
 0.54
 1
 0.54
 − 0.42
Plot the points and graph
Graph f(t) = (sin(x), √x) for 0 ≤ t ≤ 2π
• Make a table
•  t
 0
 2
 π
 [(3π)/2]
 2π
 x(t)
 0
 1
 0
 − 1
 0
 y(t)
 0
 1.25
 1.77
 2.17
 2.51
Plot the points and graph
A parametric curve is defined by
y(t) = t2
x(t) = t + 2
What is its Cartesian equation?
• Solve for t with x
• x = t + 2
• t = x − 2
Substitute into y(t)
y(t) = t2
y = (x − 2)2
A parametric curve is defined by:
y(t) = t − 9
x(t) = √t
for t ≥ 0
What is its Cartesian equation?
• Solve for t, we start with x
• x = √t
• t = x2
Substitute into y(t)
y(t) = t − 9
y = x2 − 9
Find the Cartesian equation of the following parametric equations:
x(t) = [cos(t)/3]
y(t) = [sin(t)/4]
for t ≥ 0
• Isolate x and y
• x = [cos(t)/3]
• 3x = cos(t)
• y = [sin(t)/4]
• 4y = sin(t)
Apply Pythagorean identity
cos2(t) + sin2(t) = 1
(3x)2 + (4y)2 = 1
9x2 + 16y2 = 1
Find the Cartesian equation of the following parametric equations:
x(t) = √2 sin(t)
y(t) = 2cos(t) + 3
for t ≥ 0
• Isolate x
• x = √2 sin(t)
• [x/(√2 )] = sin(t)
• Isolate y
• y = 2cos(t) + 3
• y − 3 = 2cos(t)
• [(y − 3)/2] = cos(t)
sin2(t) + cos2(t) = 1
([x/(√2 )])2 + ([(y − 3)/2])2 = 1
[(x2)/2] + [((y − 3)2)/4] = 1
Find the Cartesian equation of the following parametric equations:
x(t) = t + 5
y(t) = t3 + 5
• Isolate t with x
• x = t + 5
• x − 5 = t
y = (x − 5)3 + 125
y = x3 − 15x2 + 75x − 125 + 125
y = x3 − 15x2 + 75x
Find the Cartesian equation of the following parametric equations:
x(t) = 1 + t
y(t) = t2 − 9
• Isolate t with x
• x = 1 + t
• − t = 1 − x
• t = x − 1
Substitute into y(t)
y = (x − 1)2 − 9
y = x2 − 2x + 1 − 9
y = x2 − 2x − 8
Find the Cartesian equation of the following parametric equations, and graph it:
x(t) = t2 − 3
y(t) = 1 − t
for t − 3
• Isolate t with x
• x = t2 − 3
• x + 3 = t2
• √{x + 3} = t
• Substitute into y(t)
• y = 1 − √{x + 3}
• Graph the equation using shifts
• y = 1 − √{x + 3}
The graph √x has been shifted up by 1 unit to y = 1, and left by 3 units. It also has been reflected.
Find the Cartesian equation of the following parametric equations, and graph it:
x(t) = arccos([t/4]) − [(π)/2]
y(t) = [t/8] − 3 for − 2π ≤ t ≤ 2π
• Isolate t with x
• x = arccos([t/4]) − [(π)/2]
• x + [(π)/2] = arccos([t/4])
• cos(x + [(π)/2]) = [t/4]
• 4cos(x + [(π)/2]) = t
• Observe half - angle trig identity
• cos(u + [(π)/2]) = − sinu
• 4cos(x + [(π)/2]) = t
• − 4sinx = t
• Substitute into y(t)
• y = [( − 4sin(x))/8] − 3
• y = − [sinx/2] − 3
• Graph using shifts, reflections, and stretching
• y = − [sinx/2] − 3
• The graph y = sinx has been shifted 3 units down to y = − 3, and reflected. It also has been compressed by a factor of 2.
• Remember to graph within the bounds − 2π ≤ t ≤ 2π.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Parametric Curves

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Parametric Equations 0:23
• Familiar Functions
• Parametric Equation/ Function
• Example 1: Graph Parametric Equation 1:48
• Example 2 4:30
• Example 3 6:01
• Example 4 7:12
• Example 5 8:10