Show each step of your work and fully simplify each expression.
Turn in your answers in class on a physical piece of paper.
Staple multiple sheets together.
Feel free to use Desmos for graphing.
Answer the following:
Convert $r = \sin 2\theta$ into rectangular form. Hint: Identities.
In polar coordinates, what do the symbols $r = f(\theta)$ mean?
Using Desmos, sketch a rough graph of the following and describe it in English.
Hints: In Desmos, you can create the $\theta$ symbol by typing the letters "theta".
You can type the following out in Desmos symbol for symbol and Desmos will understand you want a polar graph.
Don't forget the parentheses!
$r = \theta$ Zoom out for this one!
$r = 3\sin\theta$
$r = 6\cos \theta$
$r = 1 + \cos \theta$
$r = \cos (1.5\theta)$
$r = \sin \left(2\theta\right)$
$r = \sin \theta + \sin\left(\dfrac{5\theta}{2}\right)$
Drag the slider $a$ from left to right to see this one drawn out!
What is the motivation for plane curves? How about for parametric equations?
For each of the following functions:
Sketch a plane curve.
Find a rectangular coordinate equation for the plane curve by eliminating the parameter.
$x = 1, \quad y = t$
$x = 2t, \quad y = t + 1$
$x = t, \quad y = t^2$
$x = t^2, \quad y = t$
$x = t^2-1, \quad y = t^2$
$x = \sqrt{t}, \quad y = t$
$x = t, \quad y = \sin \left(t - \pi\right), \quad \pi \leq t \leq 3\pi$
$x = 2\cos t, \quad y = 2\sin t, \quad 0 \leq t \leq \pi$
How do you convert a polar equation $r = f(\theta)$ into parametric form?
Express the following polar equations in parametric form:
$r = 1$
$r = \sin \theta$
$r = \cos(2\theta)$
$r = 1 + \cos \theta$
$r = \sin \theta + \cos \theta$
What is a conic section?
In algebra we expressed parabolas as $y = ax^2 + bx + c$. There is also a geometric interpretation. What is it?
What is the definition of a focus and a directrix of a parabola?
If the directrix is horizontal and the focus is the underneath the directrix, does the parabola open upwards or downwards?
If the directrix is vertical and the parabola opens to the left, is the focus to the left or right of the directrix?
For each of the following parabolas:
Find the focus, directrix, and focal diameter of the parabola.
Sketch a rough graph of the parabola, its focus and its directrix.