Algebra 2 Unit 9: Conics Overview

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Questions and Answers

What are the four shapes of graphs produced in the conics we study?

Ellipses (correct)
Hyperbolas (correct)
Parabolas (correct)
Circles (correct)

What are the characteristics that can be identified from a parabola?

focus, directrix, vertex

What are the two types of parabolas formed in conics?

horizontal parabola and vertical parabola

What is the formula for finding the vertex in both a horizontal and vertical parabola?

(h, k) Signup and view all the answers

What is the general formula/equation for a vertical parabola?

(x-h)²=4p(y-k) Signup and view all the answers

Which axis is a vertical parabola parallel to?

y-axis Signup and view all the answers

What does p represent in the general equation for a parabola?

The distance from the vertex to the focus, which is equivalent to the distance from the vertex to the directrix. Signup and view all the answers

What component of the general equation of conics is almost never found in an equation for a parabola?

Bxy Signup and view all the answers

What is the general equation for all conics?

Ax²+Bxy+Cy²+Dx+Ey+F=0 Signup and view all the answers

What is the midpoint formula?

((x₁+x₂)/2, (y₁+y₂)/2) Signup and view all the answers

What is the distance formula?

d²=(x₁-x₂)²+(y₁-y₂)² Signup and view all the answers

How can one know that a conic equation is for a parabola?

If either A or C equals zero, or if only one term is squared. Signup and view all the answers

How can using the discriminant of the quadratic equation help prove if a conic equation is a parabola?

If b²-4ac=0, the conic is possibly a parabola. Signup and view all the answers

What is the formula for finding the focus in a vertical parabola?

(h, k+p) Signup and view all the answers

What is the formula for finding the directrix in a vertical parabola?

y=k-p Signup and view all the answers

What is the general formula/equation for a horizontal parabola?

(y-k)²=4p(x-h) Signup and view all the answers

What is the formula for finding the focus in a horizontal parabola?

(h+p, k) Signup and view all the answers

What is the formula for finding the directrix in a horizontal parabola?

x=h-p Signup and view all the answers

How does one know if the p-value in a conic equation for a parabola is negative?

If the parabola opens downwards (when vertical) or opens to the left (when horizontal), the p-value is negative. Signup and view all the answers

How can one know that a conic equation is for a circle?

If A=C, then the equation is possibly a circle. Signup and view all the answers

How can using the discriminant of the quadratic equation help prove if a conic equation is a circle?

If b²-4ac=0, then the conic is possibly a hyperbola. Signup and view all the answers

How can one determine if a hyperbola is horizontal or vertical?

If x is positive, the hyperbola is horizontal; if y is positive, the hyperbola is vertical. Signup and view all the answers

What is the general equation for a horizontal hyperbola?

(x-h)²/a²-(y-k)²/b²=1 Signup and view all the answers

What are the characteristics of a horizontal hyperbola?

center, vertices, foci, asymptotes Signup and view all the answers

What is the general formula for finding the center of either a horizontal or vertical hyperbola?

(h, k) Signup and view all the answers

How can one find the two vertices of a horizontal hyperbola?

Add/subtract a value to the h value of the center. Signup and view all the answers

What is the formula for finding the values of the foci in either a vertical or horizontal hyperbola?

c²=a²+b² Signup and view all the answers

How can one find the coordinates of the foci of a horizontal hyperbola?

Add/subtract the c value from the h value of the center. Signup and view all the answers

What are the two formulas for finding the asymptotes in a horizontal hyperbola?

(y-k)=b/a(x-h) AND (y-k)=-b/a(x-h) Signup and view all the answers

What is the general equation for a vertical hyperbola?

(y-k)²/a²-(x-h)²/b²=1 Signup and view all the answers

What are the characteristics of a vertical hyperbola?

center, vertices, foci, asymptotes Signup and view all the answers

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Study Notes

Conic Sections Overview

Four primary shapes of conic graphs: Circles, Ellipses, Hyperbolas, Parabolas.

Parabola Characteristics

Key features include the focus, directrix, and vertex.
Two types exist: horizontal and vertical parabolas.

Parabola Formulas

Vertex coordinates represented as (h, k).
General formula for vertical parabolas: (x-h)²=4p(y-k).
Vertical parabolas are aligned with the y-axis.
The variable p stands for the distance from the vertex to both the focus and directrix.

Conic Equations

General equation for all conics: Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Rarely includes the Bxy term in parabola equations.
A parabola can be identified in a conic equation if either A or C equals zero, indicating only one squared term.

Discriminant Use

Discriminant b² - 4ac = 0 suggests a conic may be a parabola.

Focus and Directrix Formulas

For vertical parabolas: Focus at (h, k+p); Directrix at y = k-p.
For horizontal parabolas: General formula is (y-k)²=4p(x-h); Focus at (h+p, k); Directrix at x=h-p.

Orientation and p-value

A negative p-value indicates the parabola opens downward (vertical) or leftward (horizontal).

Circle Identification

A conic is a circle if A = C in the general equation.

Hyperbola Identification

A hyperbola is identified in standard form by the sign of the squared terms: positive for x indicates a horizontal hyperbola, while positive for y indicates a vertical hyperbola.

Hyperbola Equations

General horizontal hyperbola equation: (x-h)²/a² - (y-k)²/b² = 1.
Characteristics include center, vertices, foci, and asymptotes.
Center for hyperbolas is given by (h, k).

Focus and Vertex Calculations

Vertices computed by adding or subtracting 'a' from the h-coordinate of the center.
Foci determined using the formula c² = a² + b², and their coordinates are found by adjusting the center's h-coordinate by ±c.

Asymptotes of Hyperbola

Asymptotes for horizontal hyperbola: (y-k) = (b/a)(x-h) and (y-k) = -(b/a)(x-h).
For vertical hyperbolas, similar asymptote formulations apply, using the appropriate centered coordinates.

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