Parametric Functions Modeling Planar Motion - AP Precalculus

1. What type of curve is $x = a \cos(t)$, $y = a \sin(t)$?

   A circle. A circle centered at origin with radius $a$.

2. Find $x$ at $t = 0$ for $x = 7t - 2$.

   $x = -2$. Substitute $t = 0$: $x = 7(0) - 2 = -2$.

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

   $y^2 = 4(x + 1)$. From $y = 2t$, get $t = \frac{y}{2}$, substitute into $x$.

4. What is the path of $x = 2 \cos(t)$, $y = 3 \sin(t)$?

   An ellipse. Standard form of an ellipse with semi-axes 2 and 3.

5. What is the trajectory of $x = t$, $y = 3t + 2$?

   A line. Linear relationship between $x$ and $y$ with slope 3.

6. State the parametric equations for a circle with radius $r$.

   $x = r \, \cos(t)$, $y = r \, \sin(t)$. Standard form using cosine for $x$ and sine for $y$ components.

7. Convert $x = t$, $y = 2t + 3$ to Cartesian form.

   $y = 2x + 3$. Since $x = t$, substitute directly into $y = 2t + 3$.

8. Find the slope of the line for $x = 2t + 1$, $y = 3t - 4$.

   Slope is $\frac{dy}{dx} = \frac{3}{2}$. Slope equals $\frac{dy/dt}{dx/dt} = \frac{3}{2}$.

9. What shape does $x = t$, $y = t^2$ describe?

   A parabola. Standard parabola opening upward.

10. Convert $x = 4t$, $y = 5t + 1$ to a Cartesian equation.

    $y = \frac{5}{4}x + 1$. From $x = 4t$, get $t = \frac{x}{4}$, substitute into $y$.

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

    $y^2 = 4x$. From $x = t^2$, get $t = \pm\sqrt{x}$, substitute into $y = 2t$.

12. State the parametric form for a line parallel to $x$-axis.

    $x = t$, $y = c$. Horizontal line where $y$ remains constant.

13. Identify the parameter in the equations $x = 3t$, $y = 2t + 1$.

    The parameter is $t$. The independent variable that both $x$ and $y$ depend on.

14. Convert $x = t + 1$, $y = t^2$ to a Cartesian equation.

    $y = (x-1)^2$. From $x = t + 1$, get $t = x - 1$, substitute into $y = t^2$.

15. State the parametric form of a line through $(x_0, y_0)$ with slope $m$.

    $x = x_0 + t$, $y = y_0 + mt$. General form where $t$ acts as the parameter for direction.

16. What is the path of $x = t$, $y = t^3$?

    A cubic curve. Third-degree polynomial relationship.

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

    $y = \frac{1}{3}x + 3$. From $x = 6t$, get $t = \frac{x}{6}$, substitute into $y$.

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

    $y = \frac{2}{3}x + \frac{10}{3}$. From $x = 3t + 1$, get $t = \frac{x-1}{3}$, substitute into $y$.

19. Find $x$ at $t = 2$ for $x = 4t + 3$.

    $x = 11$. Substitute $t = 2$: $x = 4(2) + 3 = 11$.

20. What is the range of $y = 2 \sin(t)$ for $0 \leq t \leq 2\pi$?

    $-2 \leq y \leq 2$. Sine function oscillates between -1 and 1, scaled by factor 2.

21. State parametric equations for the horizontal line $y = c$.

    $x = t$, $y = c$. Let $t$ vary while keeping $y$ constant at $c$.

22. What motion does $x = 3 \cos(t)$, $y = 3 \sin(t)$ represent?

    Circular motion. Parametric equations for a circle with radius 3.

23. Find the range of $y = \sin(t)$ for $0 \leq t \leq 2\pi$.

    $-1 \leq y \leq 1$. Standard range of the sine function.

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

    $x^2 + y^2 = 1$. Uses the Pythagorean identity $\cos^2(t) + \sin^2(t) = 1$.

25. What is the meaning of $t$ in parametric equations?

    A parameter, often representing time. Usually represents time or another independent variable.

26. Find $y$ at $t = 2$ for $y = 3t - 5$.

    $y = 1$. Substitute $t = 2$: $y = 3(2) - 5 = 1$.

27. Find the coordinates at $t = 0$ for $x = t^2$, $y = 2t$.

    $(0, 0)$. Both coordinates are zero when $t = 0$.

28. Convert $x = 2t + 3$, $y = 4t - 1$ to Cartesian form.

    $y = 2x - 7$. From $x = 2t + 3$, get $t = \frac{x-3}{2}$, substitute into $y$.

29. State the parametric equations for a line with slope $m$.

    $x = x_0 + at$, $y = y_0 + mt$. General form with direction vector $(a, m)$ and slope $\frac{m}{a}$.

30. Find the point at $t = 1$ for $x = 3t$, $y = t^2 + 1$.

    $(3, 2)$. Substitute $t = 1$: $x = 3$, $y = 1 + 1 = 2$.

31. Find $y$ at $t = 3$ for $y = 2t - 1$.

    $y = 5$. Substitute $t = 3$: $y = 2(3) - 1 = 5$.

32. Find the initial point of $x = 2t + 1$, $y = 3t$ at $t = 0$.

    $(1, 0)$. Substitute $t = 0$: $x = 1$, $y = 0$.

33. State the parametric form of a vertical line $x = c$.

    $x = c$, $y = t$. Let $t$ vary while keeping $x$ constant at $c$.

34. What is the result of $x = 2t$, $y = 3t$?

    A line through the origin. Linear relationship with slope $\frac{3}{2}$ passing through origin.

35. Find $x$ at $t = 4$ for $x = 5t - 3$.

    $x = 17$. Substitute $t = 4$: $x = 5(4) - 3 = 17$.

36. What is a parametric equation?

    An equation expressing coordinates as functions of a parameter. Both $x$ and $y$ are expressed in terms of an independent variable.

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