Conics: Standard Form

Pre-CalculusConics part 1

PARABOLA

• A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.

Standard Form of a Parabola

Vertex:Axis of Symm:Opens?

Vertex:Axis of Symm:Opens?

Find the standard form of the following:

𝑥2+4 𝑥+6 𝑦−2=0

Why does this graph look like this for this equation? What are the relationships?

Find the standard form of the following:

𝑦 2+𝑥+𝑦=0

Why does this graph look like this for this equation? What are the relationships?

Circle

• A circle is the set of all points in space that are a given distance from a fixed point called the center.

Standard form of a Circle

Center:Radius:

Find the standard form:

𝑥2+𝑦2+14 𝑥−12 𝑦+4=0

Why does this graph look like this for this equation? What are the relationships?

Find the standard form:

137+6 𝑦=−𝑥2− 𝑦2−24 𝑥

Why does this graph look like this for this equation? What are the relationships?

Ellipse

The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major axis, and its midpoint is the center of the ellipse. The chord perpendicular to the major axis through the center of the ellipse is the minor axis of the ellipse. It intersects the ellipse at its co-vertices.

Standard Form of an Ellipse

Center at (h, k)Major axis is horizontal (larger value under x)

Center at (h, k)Major axis is vertical (larger value is under y)

Given 0 < b < a

Write in standard form:

𝑥2+4 𝑦2+6 𝑥−8 𝑦+9=0

Why does this graph look like this for this equation? What are the relationships?

Write in standard form:

4 𝑥2+𝑦2−8 𝑥+4 𝑦−8=0

Why does this graph look like this for this equation? What are the relationships?

HYPERBOLAA hyperbola has two disconnected branches. The line through the foci intersects the hyperbola at its two vertices and is called the transverse axis. The midpoint of the transverse axis is the center of the hyperbola.

Standard Form of a Hyperbolaa is not largest value, but first denominator

Center:Transverse axis is horizontal (x over larger value)

Center:Transverse axis is vertical (y over larger value )

Write in standard form:

9 𝑥2− 𝑦2−36 𝑥−6 𝑦+18=0

Why does this graph look like this for this equation? What are the relationships?

Write in standard form:

16 𝑦 2−𝑥2+2𝑥+64 𝑦+47=0

Why does this graph look like this for this equation? What are the relationships?

SUMMARIZE…• How can you determine what type of conic

section an equation represents from the generic formula…

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0Parabola x is squared or y is squared, but not both

Circle

Ellipse

Hyperbola …

1) 2

Classify each equation.