Parametrization of Implicitly Defined Functions - AP Precalculus

1. Identify the parametric form of the ellipse $x^2 + 4y^2 = 4$.

   $x = 2\text{cos}(t), y = \text{sin}(t)$. Ellipse in form $\frac{x^2}{4} + y^2 = 1$ uses appropriate scaling.

2. Parametrize the line $y = 5x + 7$ using $x = t$.

   $x = t, y = 5t + 7$. Linear function with slope 5 parametrized using $x = t$.

3. Parametrize $y = \frac{1}{2}x + 3$ using $x = t$.

   $x = t, y = \frac{1}{2}t + 3$. Linear equation with fractional slope parametrized using $x = t$.

4. Parametrize the line segment from $(1, 2)$ to $(3, 4)$.

   $x = 1 + 2t, y = 2 + 2t$ for $t \text{ in } [0, 1]$. Parameter $t$ ranges from 0 to 1 to connect the endpoints.

5. Express $y = 3x + 2$ parametrically.

   $x = t, y = 3t + 2$. Setting $x = t$ allows direct substitution into linear form.

6. Convert $y = x^3$ to parametric form using $x = t$.

   $x = t, y = t^3$. Direct substitution of $x = t$ into the cubic function.

7. What is the parametric form for the hyperbola $x^2 - y^2 = a^2$?

   $x = a\text{cosh}(t), y = a\text{sinh}(t)$. General hyperbola form scales functions by parameter $a$.

8. Parametrize the hyperbola $x^2 - y^2 = 1$.

   $x = \text{cosh}(t), y = \text{sinh}(t)$. Hyperbolic functions satisfy $\cosh^2(t) - \sinh^2(t) = 1$.

9. Convert $y = \frac{1}{x}$ to parametric form using $x = t$.

   $x = t, y = \frac{1}{t}$. Setting $x = t$ gives reciprocal function for $y$.

10. What is the parametric form of $x^2 - y^2 = 4$?

    $x = 2\text{cosh}(t), y = 2\text{sinh}(t)$. Hyperbola with $a = 2$ uses scaled hyperbolic functions.

11. Convert $y = x^2 - 1$ to parametric form using $x = t$.

    $x = t, y = t^2 - 1$. Parabola shifted down 1 unit from standard form.

12. Express $y = x - 4$ parametrically.

    $x = t, y = t - 4$. Linear function with slope 1 and $y$-intercept -4.

13. What is the parametric form of $x^2 + y^2 = r^2$?

    $x = r\text{cos}(t), y = r\text{sin}(t)$. General circle equation uses radius $r$ to scale unit circle.

14. Convert $y = x^3 + 2$ to parametric form using $x = t$.

    $x = t, y = t^3 + 2$. Cubic function shifted up 2 units from standard form.

15. Parametrize the parabola $y = ax^2$ using $x = t$.

    $x = t, y = at^2$. General parabolic form with parameter $a$ controlling curvature.

16. State the parametric form for the ellipse $4x^2 + 9y^2 = 36$.

    $x = 3\text{cos}(t), y = 2\text{sin}(t)$. Ellipse in standard form $\frac{x^2}{9} + \frac{y^2}{4} = 1$ uses scaled trig functions.

17. Convert $y = x^2 + 1$ to parametric form using $x = t$.

    $x = t, y = t^2 + 1$. Parabola shifted up by 1 unit uses parameter substitution.

18. Identify the parameter in $x = 2\text{cos}(t), y = 2\text{sin}(t)$.

    $t$ is the parameter. The independent variable controlling both coordinate functions.

19. Parametrize $y = 3x - 2$ using $x = t$.

    $x = t, y = 3t - 2$. Linear equation parametrized by direct substitution of $x = t$.

20. What is the parametric form of the circle $x^2 + y^2 = 16$?

    $x = 4\text{cos}(t), y = 4\text{sin}(t)$. Circle with radius 4 centered at origin uses scaled functions.

21. Convert $y = x^2$ to parametric form using $x = t$.

    $x = t, y = t^2$. Direct substitution of $x = t$ into the parabolic equation.

22. What is the parametric form of $x^2 - y^2 = 9$?

    $x = 3\text{cosh}(t), y = 3\text{sinh}(t)$. Hyperbola with $a = 3$ scales the standard hyperbolic functions.

23. What is the parametric form of the unit circle $x^2 + y^2 = 1$?

    $x = \text{cos}(t), y = \text{sin}(t)$. Uses trigonometric identities where $\cos^2(t) + \sin^2(t) = 1$.

24. Parametrize the line $y = -x + 5$ using $x = t$.

    $x = t, y = -t + 5$. Negative slope line parametrized by setting $x = t$.

25. Find the parametric equations for the circle centered at $(0,0)$ with radius $r$.

    $x = r\text{cos}(t), y = r\text{sin}(t)$. General form scales the unit circle by radius $r$.

26. Find parametric equations for $y = 2x + 1$ using $x = t$.

    $x = t, y = 2t + 1$. Direct substitution of parameter $t$ for variable $x$.

27. What parameter values form a semicircle?

    $0 \text{ to } \text{π}$ or $\text{π} \text{ to } 2\text{π}$. Half the full circle parameter range of $0$ to $2\pi$.

28. What is the parameter range for a full circle parametrization?

    $0 \text{ to } 2\text{π}$. Complete revolution around the circle requires $2\pi$ radians.

29. Parametrize the line $y = 2x + 3$ using $t$ as a parameter.

    $x = t, y = 2t + 3$. Setting $x = t$ directly substitutes into the linear equation.

30. Identify the parameter in $x = 3t + 1$, $y = 2t - 4$ for $t$ in $[0, 1]$.

    $t$ is the parameter. The variable $t$ controls the values of both $x$ and $y$ coordinates.

31. Parametrize $x^2 + y^2 = 25$.

    $x = 5\text{cos}(t), y = 5\text{sin}(t)$. Circle with radius 5 centered at origin uses scaled unit circle.

32. What is the parametric form of the ellipse $9x^2 + 4y^2 = 36$?

    $x = 2\text{cos}(t), y = 3\text{sin}(t)$. Standard form $\frac{x^2}{4} + \frac{y^2}{9} = 1$ determines scaling factors.

33. Find the parametric equations for the line through $(1, 2)$ and $(4, 6)$.

    $x = 1 + 3t, y = 2 + 4t$. Direction vector $(3,4)$ gives parametric form from point $(1,2)$.

34. Express $x^2 + y^2 = 9$ using parameter $t$.

    $x = 3\text{cos}(t), y = 3\text{sin}(t)$. Circle with radius 3 uses scaled trigonometric functions.

35. Convert $y = \frac{1}{x^2}$ to parametric form using $x = t$.

    $x = t, y = \frac{1}{t^2}$. Reciprocal squared function with parameter substitution.

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