Parabola opens left/right:

$(y - k)^2 = a(x - h)$

Flip to see [answer/question]

Revise later

SpaceTo flip

If confident

All Flashcards

Parabola opens left/right:

$(y - k)^2 = a(x - h)$

Parabola opens up/down:

$a(y - k) = (x - h)^2$

Ellipse Standard Equation:

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

Circle Equation:

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

Hyperbola (opens left/right):

$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

Hyperbola (opens up/down):

$-\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

Relationship between a, b, and c in an ellipse:

$c^2 = a^2 - b^2$

What is the formula to find the foci of a hyperbola?

$c^2 = a^2 + b^2$

What is the eccentricity of an ellipse?

$e = \frac{c}{a}$

What is the eccentricity of a hyperbola?

$e = \frac{c}{a}$

What are the key differences between an ellipse and a hyperbola?

Ellipse: '+' sign between squared terms, closed curve. | Hyperbola: '-' sign, two separate curves.

Compare and contrast the equations of a circle and an ellipse.

Circle: $(x-h)^2 + (y-k)^2 = r^2$ , equal radii. | Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ , different horizontal and vertical radii.

What are the similarities and differences between parabolas and hyperbolas?

Parabola: One focus and directrix, one squared term. | Hyperbola: Two foci, two squared terms with a minus sign.

Compare the foci of an ellipse and a hyperbola.

Ellipse: Foci are inside the curve, sum of distances to foci is constant. | Hyperbola: Foci are outside the curve, difference of distances to foci is constant.

Contrast the role of 'a' in the equations of an ellipse and a hyperbola.

Ellipse: 'a' is the semi-major axis length. | Hyperbola: 'a' is the distance from the center to the vertices along the transverse axis.

Compare the relationship between a, b, and c in ellipses and hyperbolas.

Ellipse: $c^2 = a^2 - b^2$ . | Hyperbola: $c^2 = a^2 + b^2$ .

Contrast the meaning of the center (h, k) in the context of a circle versus a hyperbola.

Circle: The center is equidistant from all points on the circle. | Hyperbola: The center is the midpoint between the vertices and foci, but not on the hyperbola itself.

Compare the use of asymptotes in hyperbolas versus ellipses.

Hyperbola: Has asymptotes that guide the branches of the curve. | Ellipse: Does not have asymptotes.

Contrast the way the orientation of a conic section is determined for parabolas versus hyperbolas.

Parabola: Determined by which variable is squared and the sign of the leading coefficient. | Hyperbola: Determined by which term (x or y) is positive in the standard equation.

Compare the standard form equations of ellipses and hyperbolas centered at the origin.

Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ . | Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ .

What is a conic section?

Shapes formed when a plane intersects a cone.

Define a parabola.

All points equidistant from a focus and a directrix.

What is the vertex of a parabola?

The turning point of the parabola.

Define an ellipse.

A conic section where the sum of the distances from any point on the ellipse to two foci is constant.

What are the foci of an ellipse?

Two special points inside the ellipse used in its definition.

Define a hyperbola.

A conic section formed by the intersection of a plane with a double cone.

What are the asymptotes of a hyperbola?

Lines that the hyperbola approaches as it extends to infinity.

What is the directrix of a parabola?

A line such that every point on the parabola is equidistant from the focus and the directrix.

What is the major axis of an ellipse?

The longest diameter of the ellipse, passing through the center and both foci.

What is the minor axis of an ellipse?

The shortest diameter of the ellipse, passing through the center and perpendicular to the major axis.