Precalculus

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Rules of Thumb

Conic sections are sections, or slices, of a cone
There are more than 4 conic sections. Don't forget degenerate cases (point, straight line, intersecting lines)
All conic section formulas can be written in the form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$
Although we often think of some conic sections (parabolas) as functions, that is not necessarily always the case. While it is true that these can be expressed as a polynomial of a 2nd degree, this does not mean that all parabolas are as such.


What to Review

Brush up on some geometry and graph behavior (e.g. domain and range, end behavior, other graph properties)
Quadratic formula is useful
Brush up on the following from algebra II: focus, directrix, discriminant (remember: if discriminant is positive, then the conic is an ellipse, circle, or point; if 0, then a parabola or straight line; if negative then hyperbola or intersecting lines)
Don't forget basic skills such as factoring and completing the square


Circles

Equation: $(x-h)^2+(y-k)^2=r^2$
$(h,k)$ is the center
Formulas
Area: $A=\pi r^2$
Circumference: $C=2\pi r$
Arc length: $S=r\theta_R$ or $S=\frac{\theta}{360}(2\pi r)$


Ellipses

Equation: $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$
$(h,k)$ is center, circle is special case of ellipse
Wherever the bigger denominator is is which direction the ellipse is stretched out in (ex. if $\frac{x^2}{100}+\frac{y^2}{64}=1$, the ellipse would be outstretched in the $x$ direction.


Parabolas

Equations (“Vertical Parabolas”)
Standard form: $y=ax^2+bx+c$
Vertex form: $y=a(x-h)^2+k$; vertex can also be found from standard form by using $(\frac{-b}{2a},f(\frac{-b}{2a}))$
Conic form: $4p(y-k)=(x-h)^2$; $p$ is distance from vertex to focus
Directrix form (not common): $y=\frac{1}{2(b-k)}(x-a)^2+\frac{(b-k)}{2}; (a,b)$ are coordinates of focus and $k$ y-value is directrix (line). A parabola is defined as having all points equidistant from its focus and directrix.
“Horizontal Parabolas”
Standard form: $x=ay^2+by+c$
Vertex form: $x=a(y-k)^2+h$
Conic form: $4p(x-h)=(y-k)^2$


Hyperbolas

“Horizontal” $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$
$(h,k)$ =center
$a$= distance from center to any vertex
$c$= distance from center to foci
To find this, use Pythagorean Theorem ($a^2+b^2=c^2$)
Asymptotes: $y-k=\pm \frac{b}{a}(x-h)$
End Behavior: goes to infinity as $x\to\infty$ and as $x\to-\infty$ except between asymptotes.
“Vertical” $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$
$(h,k)$ =center
$a$= distance from center to any vertex
$c$= distance from center to foci
To find this, use Pythagorean Theorem ($a^2+b^2=c^2$)
Asymptotes: $y-k=\pm\frac{a}{b}(x-h)$
End Behavior: goes to infinity as $x\to\infty$ and as $x\to-\infty$