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Parametrization of Implicitly Defined Functions

Olivia King

Olivia King

7 min read

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Study Guide Overview

This study guide covers parametrization of implicit functions, focusing on how to represent curves using parametric equations x(t) and y(t). It explains the general concept of parametrization and provides examples, including the unit circle. The guide details parametrization techniques for functions, inverse functions, parabolas, ellipses, and hyperbolas, emphasizing key formulas and potential pitfalls. It also includes practice questions and exam tips covering common question types, time management strategies, and important reminders for the final exam.

AP Pre-Calculus: Parametrization of Implicit Functions - The Night Before ๐ŸŒƒ

Hey, future AP Calc master! Let's make sure you're totally prepped for tomorrow's exam. We're diving into parametrizing implicitly defined functions, which sounds fancy but is totally doable. Think of it as giving curves a voice using a parameter, usually 't'. Let's get started!

Parametrization: Giving Curves a Voice ๐Ÿ—ฃ๏ธ

What's Parametrization?

  • Parametrization is like giving a curve a set of coordinates that depend on a single variable, 't'. Instead of just x and y, we have x(t) and y(t). These equations describe the curve as 't' changes.

  • Think of it like a GPS: 't' tells you where you are on the curve at any given moment. ๐Ÿ“

Key Concept

The key idea is that when you plug x(t) and y(t) into the original equation of the curve, it should always be true for all values of 't' in the domain. This is how you know you have a valid parametrization.

Example: The Unit Circle ๐ŸŽฏ

  • The equation xยฒ + yยฒ = 1 describes a circle. A parametrization is:
    • x(t) = cos(t)
    • y(t) = sin(t)
    • Domain: 0 โ‰ค t < 2ฯ€
  • Substitute these into the original equation:
    • cosยฒ(t) + sinยฒ(t) = 1 (which is always true!)

Parametrizing Functions: y = f(x) ๐Ÿ“ˆ

  • If you have a function y = f(x), you can parametrize it as:
    • x(t) = t
    • y(t) = f(t)
  • Imagine 't' moving along the x-axis, and f(t) gives you the corresponding y value on the graph. It's like tracing the curve with a pen! โœ๏ธ

Parametrizing Inverse Functions: x = fโปยน(y) โช

  • If f has an inverse, fโปยน, you can parametrize it as:

    • x(t) = fโปยน(t)
    • y(t) = t
  • Here, 't' moves along the y-axis, and fโปยน(t) gives you the corresponding x value on the graph. It's like tracing the curve from a different perspective! ๐Ÿ”„

Exam Tip

Remember: The domain of 't' for the inverse function is the range of the original function. Pay attention to the limits!

Parametrizing Conic Sections ๐Ÿ‘‘

Parabolas โ†ช๏ธ

  • Parabolas can be parametrized by solving for one variable and expressing the other in terms of 't'.
  • If you can solve for x: x(t) = f(t), y(t) = t
  • If you can solve for y: x(t) = t, y(t) = f(t)

Parametric and rectangular parabolas

Parametric and rectangular parabolas

Common Mistake

Don't mix up which variable is 't'! If x is solved in terms of y, then y=t, and vice versa.

Ellipses ๐Ÿฅš

  • Parametric equations for an ellipse centered at (h, k):
    • x(t) = h + a cos(t)
    • y(t) = k + b sin(t)
    • 'a' and 'b' are the semi-major and semi-minor axes.
    • Domain: 0 โ‰ค t โ‰ค 2ฯ€ (one full revolution).

Ellipse

Ellipse

Memory Aid

Think of it like a circle stretched in two directions. 'a' and 'b' control the stretching along the x and y axes.

Hyperbolas ๐Ÿ”

  • Horizontal Hyperbola (opens left and right):
    • x(t) = h + a sec(t)
    • y(t) = k + b tan(t)
    • Domain: 0 โ‰ค t < 2ฯ€
  • Vertical Hyperbola (opens up and down):
    • x(t) = h + a tan(t)
    • y(t) = k + b sec(t)
    • Domain: 0 โ‰ค t < 2ฯ€

Hyperbola

Hyperbola

Quick Fact

Remember: sec(t) = 1/cos(t) and tan(t) = sin(t)/cos(t). These are your friends for hyperbolas!

Final Exam Focus ๐ŸŽฏ

  • Key Topics: Parametrization of circles, ellipses, parabolas, and hyperbolas.
  • Common Questions: Finding parametric equations given an implicit equation, understanding the domain of 't', and using parametrizations to solve problems.
  • Time Management: Practice quickly identifying the type of conic section and applying the correct parametric equations. Don't spend too long on one question!
  • Common Pitfalls: Mixing up the formulas for horizontal vs. vertical hyperbolas, and not paying attention to the domain of 't'.

Practice Question

Multiple Choice Questions

  1. Which of the following is a correct parametrization for the ellipse given by the equation (xโˆ’2)29+(y+1)216=1\frac{(x-2)^2}{9} + \frac{(y+1)^2}{16} = 1? (A) x(t)=2+3cosโก(t),y(t)=โˆ’1+4sinโก(t)x(t) = 2 + 3\cos(t), y(t) = -1 + 4\sin(t) (B) x(t)=2+9cosโก(t),y(t)=โˆ’1+16sinโก(t)x(t) = 2 + 9\cos(t), y(t) = -1 + 16\sin(t) (C) x(t)=3cosโก(t),y(t)=4sinโก(t)x(t) = 3\cos(t), y(t) = 4\sin(t) (D) x(t)=2+4cosโก(t),y(t)=โˆ’1+3sinโก(t)x(t) = 2 + 4\cos(t), y(t) = -1 + 3\sin(t)

  2. A hyperbola is given by the equation y225โˆ’x29=1\frac{y^2}{25} - \frac{x^2}{9} = 1. Which of the following represents a valid parametrization for this hyperbola? (A) x(t)=3tanโก(t),y(t)=5secโก(t)x(t) = 3\tan(t), y(t) = 5\sec(t) (B) x(t)=3secโก(t),y(t)=5tanโก(t)x(t) = 3\sec(t), y(t) = 5\tan(t) (C) x(t)=5tanโก(t),y(t)=3secโก(t)x(t) = 5\tan(t), y(t) = 3\sec(t) (D) x(t)=5secโก(t),y(t)=3tanโก(t)x(t) = 5\sec(t), y(t) = 3\tan(t)

Free Response Question

Consider the parabola given by the equation y=x2โˆ’4x+7y = x^2 - 4x + 7.

(a) Find a parametrization for this parabola using the parameter tt.

(b) Find the vertex of the parabola.

(c) Determine the range of the parabola.

(d) If the parabola is parametrized as x(t)=tx(t) = t, what is the expression for y(t)y(t)?

Scoring Rubric:

(a) 1 point: Correct parametrization. x(t)=t,y(t)=t2โˆ’4t+7x(t)=t, y(t)=t^2-4t+7

(b) 2 points: 1 point for completing the square or using the vertex formula, and 1 point for identifying the vertex coordinates. Vertex: (2, 3)

(c) 1 point: Correct range. Range: [3,โˆž)[3, \infty)

(d) 1 point: Correct expression for y(t)y(t). y(t)=t2โˆ’4t+7y(t) = t^2 - 4t + 7

Last-Minute Tips ๐Ÿ’ก

  • Stay Calm: You've got this! Take deep breaths and trust your preparation.
  • Read Carefully: Pay close attention to the question's wording and what it's asking.
  • Show Your Work: Even if you make a mistake, partial credit is your friend.
  • Double-Check: If you have time, go back and review your answers.

You're ready to rock this exam! Go get 'em! ๐Ÿ’ช

Question 1 of 13

What does parametrization essentially do to a curve? ๐Ÿค”

It simplifies the equation of the curve

It gives the curve a set of coordinates that depend on a single variable, 't'

It rotates the curve by 90 degrees

It stretches or compresses the curve