Math Placement Exam - Conic Sections

Conic Sections

Consists of four geometric shapes (parabolas, circles, ellipses, and hyperbolas) that are actually the cross-sections of a right circular cone sliced by a plane

If A equals B

It's a circle

If A doesn't equal B (same signs)

It's an ellipse

If A doesn't equal B (different signs)

It's a hyperbola

General Form

A x2 + B y2 + C x+ D y + E = 0

Parabola

Is the set of points on the coordinate plane that are equidistant from a fixed point (focus) and fixed line (directrix)

Parabola's Standard Form (x2)

y = a ( x - h) 2 + k

Parabola's Standard Form (x2): vertex

( h , k )

Parabola's Standard Form (x2): focus

( h , k + c )

Parabola's Standard Form (x2): axis of symmetry

( x = h )

Parabola's Standard Form (x2): directrix

( y = k - c )

Axis of Symmetry

A line that cuts through the middle of the parabola, intersecting at the vertex

Vertex

Parabola's lowest point if graphs points up, highest point if graph points down

The Value C

The distance from the vertex to both the focus and directrix (measured along axis of symmetry); will always be POSITIVE

The Value A in the Standard Form

+/- (1/(4c))

Parabola's Standard Form (y2)

x = a ( y - k ) 2 + h

Parabola's Standard Form (y2) : Directrix

x = h - c

Parabola's Standard Form (y2): Focus

( h + c, k )

Parabola's Standard Form (y2): Axis of Symmetry

y = k

Circle

Is a set of points in the coordinate plane that are all the same distance (called the radius) from a fixed point (called the center)

Circle's Standard Form

( x - h ) 2 + (y - k ) 2 = r 2