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

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 🗣️

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. 📍

Example: The Unit Circle 🎯

• The equation x² + y² = 1 describes a circle. A parametrization is:
  • x(t) = cos(t)
  • y(t) = sin(t)
  • Domain: 0 ≤ t < 2π
• Substitute these into the original equation:
  • cos²(t) + sin²(t) = 1 (which is always true!)

Parametrizing Functions: y = f(x) 📈

• 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! ✍️

Parametrizing Inverse Functions: x = f⁻¹(y) ⏪

• If f has an inverse, f⁻¹, you can parametrize it as:

  • x(t) = f⁻¹(t)
  • y(t) = t

• Here, 't' moves along the y-axis, and f⁻¹(t) gives you the corresponding x value on the graph. It's like tracing the curve from a different perspective! 🔄

Parametrizing Conic Sections 👑

Parabolas ↪️

• 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

Ellipses 🥚

• 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 ≤ t ≤ 2π (one full revolution).

Ellipse

Hyperbolas 🔁

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

Hyperbola

Final Exam Focus 🎯

• 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 $\frac{(x-2)^2}{9} + \frac{(y+1)^2}{16} = 1$? (A) $x(t) = 2 + 3\cos(t), y(t) = -1 + 4\sin(t)$ (B) $x(t) = 2 + 9\cos(t), y(t) = -1 + 16\sin(t)$ (C) $x(t) = 3\cos(t), y(t) = 4\sin(t)$ (D) $x(t) = 2 + 4\cos(t), y(t) = -1 + 3\sin(t)$

2. A hyperbola is given by the equation $\frac{y^2}{25} - \frac{x^2}{9} = 1$. Which of the following represents a valid parametrization for this hyperbola? (A) $x(t) = 3\tan(t), y(t) = 5\sec(t)$ (B) $x(t) = 3\sec(t), y(t) = 5\tan(t)$ (C) $x(t) = 5\tan(t), y(t) = 3\sec(t)$ (D) $x(t) = 5\sec(t), y(t) = 3\tan(t)$

Free Response Question

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

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

(b) Find the vertex of the parabola.

(c) Determine the range of the parabola.

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

Scoring Rubric:

(a) 1 point: Correct parametrization. $x(t)=t, y(t)=t^2-4t+7$

(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: $[3, \infty)$

(d) 1 point: Correct expression for $y(t)$. $y(t) = t^2 - 4t + 7$

Last-Minute Tips 💡

• 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! 💪