Questions and Answers
What are the four shapes of graphs produced in the conics we study?
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?
Signup and view all the answers
What is the general formula/equation for a vertical parabola?
Signup and view all the answers
Which axis is a vertical parabola parallel to?
Signup and view all the answers
What does p represent in the general equation for a parabola?
Signup and view all the answers
What component of the general equation of conics is almost never found in an equation for a parabola?
Signup and view all the answers
What is the general equation for all conics?
Signup and view all the answers
What is the midpoint formula?
Signup and view all the answers
What is the distance formula?
Signup and view all the answers
How can one know that a conic equation is for a parabola?
Signup and view all the answers
How can using the discriminant of the quadratic equation help prove if a conic equation is a parabola?
Signup and view all the answers
What is the formula for finding the focus in a vertical parabola?
Signup and view all the answers
What is the formula for finding the directrix in a vertical parabola?
Signup and view all the answers
What is the general formula/equation for a horizontal parabola?
Signup and view all the answers
What is the formula for finding the focus in a horizontal parabola?
Signup and view all the answers
What is the formula for finding the directrix in a horizontal parabola?
Signup and view all the answers
How does one know if the p-value in a conic equation for a parabola is negative?
Signup and view all the answers
How can one know that a conic equation is for 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?
Signup and view all the answers
How can one determine if a hyperbola is horizontal or vertical?
Signup and view all the answers
What is the general equation for a horizontal hyperbola?
Signup and view all the answers
What are the characteristics of a horizontal hyperbola?
Signup and view all the answers
What is the general formula for finding the center of either a horizontal or vertical hyperbola?
Signup and view all the answers
How can one find the two vertices of a horizontal hyperbola?
Signup and view all the answers
What is the formula for finding the values of the foci in either a vertical or horizontal hyperbola?
Signup and view all the answers
How can one find the coordinates of the foci of a horizontal hyperbola?
Signup and view all the answers
What are the two formulas for finding the asymptotes in a horizontal hyperbola?
Signup and view all the answers
What is the general equation for a vertical hyperbola?
Signup and view all the answers
What are the characteristics of a vertical hyperbola?
Signup and view all the answers
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Explore the characteristics and formulas related to conic sections in Algebra 2 Unit 9. This quiz covers essential concepts such as the shapes of conic graphs, properties of parabolas, and more. Perfect for reinforcing your understanding of this crucial topic in algebra.