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Algebra 2 Unit 9: Conics Overview
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Algebra 2 Unit 9: Conics Overview

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@BeneficentHonor6192

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?

    <p>(h, k)</p> Signup and view all the answers

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

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

    Which axis is a vertical parabola parallel to?

    <p>y-axis</p> Signup and view all the answers

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

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

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

    <p>Bxy</p> Signup and view all the answers

    What is the general equation for all conics?

    <p>Ax²+Bxy+Cy²+Dx+Ey+F=0</p> Signup and view all the answers

    What is the midpoint formula?

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

    What is the distance formula?

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

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

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

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

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

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

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

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

    <p>y=k-p</p> Signup and view all the answers

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

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

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

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

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

    <p>x=h-p</p> Signup and view all the answers

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

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

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

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

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

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

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

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

    What is the general equation for a horizontal hyperbola?

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

    What are the characteristics of a horizontal hyperbola?

    <p>center, vertices, foci, asymptotes</p> Signup and view all the answers

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

    <p>(h, k)</p> Signup and view all the answers

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

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

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

    <p>c²=a²+b²</p> Signup and view all the answers

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

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

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

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

    What is the general equation for a vertical hyperbola?

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

    What are the characteristics of a vertical hyperbola?

    <p>center, vertices, foci, asymptotes</p> 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.

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    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.

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