Precalculus Analytic Geometry and Polar Coordinates Flashcards

24 sample cards from the Analytic Geometry and Polar Coordinates deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. How do you plot a parametric curve by hand?

   Make a table of values: choose several values of $t$, compute $x=f(t)$ and $y=g(t)$ for each, plot the points $(x,y)$, and connect them in order of increasing $t$ (indicating orientation with arrows).

2. What does the orientation (direction) of a parametric curve indicate?

   The direction in which the point $(x(t), y(t))$ moves along the curve as the parameter $t$ increases.

3. What does 'eliminating the parameter' mean, and why do it?

   It means combining the parametric equations into a single rectangular equation in $x$ and $y$ (removing $t$), to recognize the curve's shape as a familiar rectangular graph.

4. Describe the substitution method for eliminating the parameter.

   Solve one parametric equation for $t$ in terms of $x$ (or $y$), then substitute that expression into the other equation to get a relation between $x$ and $y$.

5. How can the Pythagorean identity eliminate the parameter from $x = a\cos t$, $y = a\sin t$?

   Since $\cos t = \frac{x}{a}$ and $\sin t = \frac{y}{a}$, use $\cos^{2} t + \sin^{2} t = 1$ to get $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2}} = 1$, i.e. the circle $x^{2}+y^{2}=a^{2}$.

6. When eliminating the parameter, why must you check the domain/range of the resulting rectangular equation?

   The parametric equations may only trace part of the full rectangular graph, so you must restrict $x$ and/or $y$ to the values actually produced by $t$.

7. What are the standard parametric equations modeling projectile motion (launch from origin, speed $v_{0}$, angle $\theta$)?

   $x = (v_{0}\cos\theta)\,t$ and $y = (v_{0}\sin\theta)\,t - \frac{1}{2}g t^{2}$, where $g$ is gravitational acceleration and $t$ is time.

8. In the projectile model, what does the horizontal equation $x = (v_{0}\cos\theta)t$ imply about horizontal motion?

   Horizontal velocity $v_{0}\cos\theta$ is constant (no horizontal acceleration), so horizontal displacement grows linearly with time.

9. In the projectile model with initial height $h$, how is the vertical equation modified?

   $y = h + (v_{0}\sin\theta)\,t - \frac{1}{2}g t^{2}$, adding the initial height $h$ to the vertical position.

10. How do you find when a projectile hits the ground using the parametric model?

    Set the vertical equation $y = 0$ and solve the resulting quadratic for $t$; the positive root is the time of impact, then substitute into $x(t)$ for the range.

11. What shape is the trajectory of an ideal projectile, and why?

    A parabola: eliminating $t$ from the linear $x(t)$ and quadratic $y(t)$ gives $y$ as a quadratic (downward-opening) function of $x$.

12. Describe the polar coordinate system.

    Each point is located by $(r, \theta)$, where $r$ is the directed distance from the origin (pole) and $\theta$ is the angle measured counterclockwise from the polar axis (positive $x$-axis).

13. What does a negative value of $r$ mean in polar coordinates?

    The point is plotted at distance $|r|$ from the pole in the direction opposite to $\theta$, i.e. along the ray at angle $\theta + \pi$.

14. Why are polar coordinates of a point not unique?

    Adding any multiple of $2\pi$ to $\theta$, or negating $r$ while adding $\pi$ to $\theta$, gives the same point; e.g. $(r,\theta)=(r,\theta+2\pi n)=(-r,\theta+\pi)$.

15. What are the formulas to convert polar coordinates $(r,\theta)$ to rectangular coordinates $(x,y)$?

    $x = r\cos\theta$ and $y = r\sin\theta$.

16. What formulas convert rectangular coordinates $(x,y)$ to polar coordinates $(r,\theta)$?

    $r = \sqrt{x^{2}+y^{2}}$ and $\tan\theta = \frac{y}{x}$ (choosing $\theta$ in the correct quadrant for $(x,y)$).

17. When converting rectangular to polar, why must you check the quadrant of $(x,y)$?

    Because $\tan\theta = \frac{y}{x}$ has the same value in two quadrants; you must select $\theta$ so that $(r\cos\theta, r\sin\theta)$ lands in the actual quadrant of the point.

18. How do you convert the rectangular equation $x^{2}+y^{2}=a^{2}$ to polar form?

    Since $x^{2}+y^{2}=r^{2}$, it becomes $r^{2}=a^{2}$, i.e. $r = a$ (a circle of radius $a$ centered at the pole).

19. How do you convert a polar equation to rectangular form in general?

    Substitute $r\cos\theta = x$, $r\sin\theta = y$, $r^{2}=x^{2}+y^{2}$, and $\tan\theta = \frac{y}{x}$; often multiply through by $r$ first to create these expressions.

20. What curve is the polar equation $r = a\cos\theta$ (or $r = a\sin\theta$)?

    A circle of diameter $a$ passing through the pole; $r=a\cos\theta$ is centered on the $x$-axis, $r=a\sin\theta$ on the $y$-axis.

21. What type of polar curve is $r = a \pm b\cos\theta$ or $r = a \pm b\sin\theta$ called, and what determines its shape?

    A limaçon. If $\frac{a}{b}<1$ it has an inner loop, if $\frac{a}{b}=1$ it is a cardioid, and if $\frac{a}{b}\geq 2$ it is convex (dimpled for $1<\frac{a}{b}<2$).

22. What is a cardioid in polar form?

    A heart-shaped limaçon where $a = b$, e.g. $r = a(1 + \cos\theta)$; it passes through the pole and has $|a|+|b|$ as its maximum $r$.

23. For the rose curve $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$, how many petals does it have?

    If $n$ is odd, there are $n$ petals; if $n$ is even, there are $2n$ petals. Each petal has length $|a|$.

24. What curve does the polar equation $r^{2} = a^{2}\cos(2\theta)$ (or $\sin 2\theta$) represent?

    A lemniscate, a figure-eight (infinity) shaped curve centered at the pole.

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Analytic Geometry and Polar Coordinates flashcards FAQ