Precalculus Analytic Geometry and Polar Coordinates Syllabus

What algebraic technique converts a general conic equation into standard (shifted) form?

Completing the square in $x$ and/or $y$ to produce perfect-square terms $(x-h)^{2}$ and $(y-k)^{2}$.

For the general second-degree equation $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$ with $B = 0$, how do you classify a parabola?

It is a parabola when exactly one of $A$ or $C$ is zero (i.e., $A \cdot C = 0$ but not both), so only one squared term appears.

For $Ax^{2} + Cy^{2} + Dx + Ey + F = 0$ (with $B=0$), how do you tell an ellipse from a hyperbola?

If $A$ and $C$ have the same sign (and $A \neq C$), it is an ellipse (a circle if $A = C$); if $A$ and $C$ have opposite signs, it is a hyperbola.

What is the discriminant test for classifying a general conic $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$?

Compute $B^{2} - 4AC$: if $< 0$ it is an ellipse (or circle), if $= 0$ a parabola, if $> 0$ a hyperbola.

What condition on $A$ and $C$ makes the conic $Ax^{2} + Cy^{2} + \dots = 0$ a circle?

$A = C$ (with the same nonzero sign); then the equation reduces to a circle.

What are parametric equations of a curve?

A pair of equations $x = f(t)$ and $y = g(t)$ expressing the coordinates as functions of a parameter $t$, tracing the curve as $t$ varies.

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).

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.

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.

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 $y$.

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 $x^{2}+y^{2}=a^{2}$.

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 $t$.

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.

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.

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.

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.

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 $x$.

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).

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 $\theta + \pi$.

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. $(r,\theta)=(r,\theta+2\pi n)=(-r,\theta+\pi)$.

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

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

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 $(x,y)$).

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.

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).

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.

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.

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 $1<\frac{a}{b}<2$).

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 $r$.

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 $|a|$.

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.