Parametric Functions - AP Precalculus

1. Determine the Cartesian equation from $x = 4 \cos(t)$, $y = 5 \sin(t)$.

   $\frac{x^2}{16} + \frac{y^2}{25} = 1$. Ellipse with semi-axes 4 and 5.

2. What is the parameter in parametric equations?

   A variable, often $t$, that both $x$ and $y$ are functions of. The independent variable that controls both coordinates.

3. Convert $x = 1 + 2t$, $y = 3 + 4t$ to a Cartesian equation.

   $y = 2x + 1$. From $x = 1 + 2t$ get $t = \frac{x-1}{2}$, substitute.

4. Convert $x = \cos(t)$, $y = \sin(t)$ to a Cartesian equation.

   $x^2 + y^2 = 1$. Unit circle using fundamental trigonometric identity.

5. What is the purpose of parametric equations in mathematics?

   To describe geometric figures and motions. Enables modeling of complex paths and trajectories.

6. Convert $x = 5t - 2$, $y = 3t + 1$ to a Cartesian equation.

   $y = \frac{3}{5}(x + 2) - 1$. From $x = 5t - 2$ get $t = \frac{x+2}{5}$, substitute.

7. What is a parametric equation?

   An equation that expresses variables as functions of a parameter. Both $x$ and $y$ depend on the same parameter.

8. Identify parametric equations for a parabola $y = x^2$.

   $x = t$, $y = t^2$. Simplest parametrization using $t$ as $x$-coordinate.

9. Identify parametric equations for a horizontal line $y = c$.

   $x = t$, $y = c$. Parameter varies while $y$ remains constant.

10. Define what a parameter is in terms of parametric equations.

    An independent variable that defines a set of equations. The controlling variable in parametric representation.

11. What is the parametric equation for a line with slope $m$?

    $x = t$, $y = mt + c$. Standard form with slope $m$ and parameter $t$.

12. Convert the parametric equations $x = t + 1$, $y = 2t$ to Cartesian form.

    $y = 2(x - 1)$. Solve for $t$ from first equation, substitute into second.

13. What is the parametric equation for a vertical line $x = c$?

    $x = c$, $y = t$. Parameter varies while $x$ remains constant.

14. Which parametric equations describe the line segment from $(1,2)$ to $(4,8)$?

    $x = 1 + 3t$, $y = 2 + 6t$, $0 \leq t \leq 1$. Direction vector $(3,6)$ with parameter range $[0,1]$.

15. What is the parametric form for a parabola $y = ax^2$?

    $x = t$, $y = at^2$. General parabola form with coefficient $a$.

16. Convert $x = 4t$, $y = 9 - t^2$ to a Cartesian equation.

    $y = 9 - \left(\frac{x}{4}\right)^2$. Substitute $t = \frac{x}{4}$ into the $y$ equation.

17. What is the role of $t$ in parametric equations?

    It is the independent variable or parameter. Controls the position along the curve as it varies.

18. Identify the parametric form for the line $y = 3x + 2$.

    $x = t$, $y = 3t + 2$. Set parameter $t = x$ for simplest form.

19. State the parametric equations for a circle centered at the origin.

    $x = r \, \cos(t)$, $y = r \, \sin(t)$. Standard form using trigonometric functions with radius $r$.

20. Convert $x = 3t$, $y = 4t$ to a Cartesian equation.

    $y = \frac{4}{3}x$. Eliminate $t$ by solving $x = 3t$ gives $t = \frac{x}{3}$.

21. Find the Cartesian equation from $x = 1 + t$, $y = 2t - 1$.

    $y = 2x - 3$. From $x = 1 + t$, get $t = x - 1$, substitute.

22. Find the Cartesian equation from $x = 2 \cos(t)$, $y = 3 \sin(t)$.

    $\frac{x^2}{4} + \frac{y^2}{9} = 1$. Use identity $\cos^2(t) + \sin^2(t) = 1$.

23. Convert $x = 3 \sin(t)$, $y = 4 \cos(t)$ to a Cartesian equation.

    $\frac{x^2}{9} + \frac{y^2}{16} = 1$. Ellipse using sine for $x$ and cosine for $y$.

24. What do $a$ and $b$ represent in the ellipse parametric equations $x = a \cos(t)$, $y = b \sin(t)$?

    The semi-major and semi-minor axes. The lengths of the ellipse's major and minor axes.

25. What is an advantage of parametric equations?

    They allow the description of curves not functions. Can represent curves that fail the vertical line test.

26. What is the parametric form of a line segment from $(2,1)$ to $(5,4)$?

    $x = 2 + 3t$, $y = 1 + 3t$, $0 \leq t \leq 1$. Direction vector $(3,3)$ from start to end point.

27. What are the parametric equations for a line parallel to $y = 2x + 3$?

    $x = t$, $y = 2t + c$. Same slope but different $y$-intercept constant.

28. Convert $x = 2 \cos(t)$, $y = 2 \sin(t)$ to a Cartesian equation.

    $x^2 + y^2 = 4$. Circle with radius 2 centered at origin.

29. Convert $x = 2t$, $y = t^2$ to a Cartesian equation.

    $y = \frac{x^2}{4}$. Eliminate parameter: $t = \frac{x}{2}$, so $y = \left(\frac{x}{2}\right)^2$.

30. What is the parametric form for a circle with radius $r$?

    $x = r \cos(t)$, $y = r \sin(t)$. General circle equation with specified radius.

31. What is the general form of parametric equations for a line?

    $x = x_0 + at$, $y = y_0 + bt$. Point $(x_0, y_0)$ with direction vector $(a,b)$.

32. Convert $x = 5 \sin(t)$, $y = 5 \cos(t)$ to a Cartesian equation.

    $x^2 + y^2 = 25$. Apply Pythagorean identity to eliminate parameter.

33. What is the parametric equation for a straight line through $(x_1, y_1)$, $(x_2, y_2)$?

    $x = x_1 + (x_2 - x_1)t$, $y = y_1 + (y_2 - y_1)t$. Linear interpolation between two given points.

34. State the parametric equations for an ellipse centered at the origin.

    $x = a \, \cos(t)$, $y = b \, \sin(t)$. Uses different coefficients for horizontal and vertical radii.

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