Lesson Conic Sections-(parabola, circle, ellipse, hyperbola)
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Algebra: Conic sections - ellipse, parabola, hyperbola
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~Parabola~ Parabolas are 'U' shaped graphed lines that have a degree of two. Standard form: {{{y = ax^2 + bx + c}}} Vertex:((-b/2a),f(x)) Zeros (the point where the parabola intersects with the x-axis) can be complex or real values. The formula (quadratic formula): {{{x = (-b +- sqrt(b^2 - 4ac))/2a}}} The discriminant is {{{b^2 - 4ac}}}. If the discriminant is greater than zero, there are two real answers. If the discriminant is equal to zero, there is one real answer. If the discriminant is less than zero, there are two complex values. Sometimes, parabolas are seen as in vertex form. Standard form: {{{y = a(x-h)^2 + k}}} Vertex: (h,k) If the {{{a}}}-value is negative, the parabola opens down. Vice versa if the {{{a}}}-value is positive. This idea works for all types of parabolas. Also, parabolas can go left and right. Standard form: {{{x = a(y-k)^2 + h}}} Vertex: (h,k) If the {{{a}}}-value is positive, the parabola opes right. Vice versa if the {{{a}}}-value is negative. ~Circle~ Standard form: {{{(x - h)^2 + (y - k)^2 = r^2}}} Center: (h,k) In the form above, {{{r}}} is the value for the radius. Formula for area of circle: {{{A=(pi)r^2}}} ~Ellipse~ Standard form: {{{((x - h)^2/a^2)+((y - k)^2/b^2)=1}}} Center: (h,k) Formula for foci(c): {{{c^2=a^2-b^2}}} The foci is (c) amount of units along the major axis. In this form, the length of the major axis (the longer axis) is {{{2a}}}. As for the length of the minor axis (the smaller axis), it is {{{2b}}}. In this form, the ellipse is horizontal because the {{{a^2}}} is under the x-values. The {{{a^2}}} is always the higher number when compared to {{{b^2}}}. If the {{{a^2}}} was under the y-values, the ellipse's major axis would be vertical. Formula for area of ellipse: {{{A=(pi)AB}}} ~Hyperbola~ Standard form: {{{((x - h)^2/a^2) - ((y - k)^2/b^2)=1}}} The {{{a^2}}} is always the value under the positive term. When the {{{a^2}}} is under the x-values, the hyperbola has a horizontal tranverse axis and the slope of its asymptotes is {{{b/a}}}. When the y-value is over the {{{a^2}}}, the transverse axis is vertical and its asymptote's slope is {{{a/b}}}. {{{2a}}} determines the transvers axis. The hyperbola is two parabolas going opposite directions that're always approaching the asymptotes, but never touching or crossing over them. Formula for foci(c): {{{c^2 = a^2 + b^2}}} The foci is located {{{c-a}}} units inside the two "parabola-like" branches. The center for the hyperbola is determine by: (h,k) In both Hyperbolas and Ellipses, the x-values are always associated with {{{h}}} and the y-values are always associated with {{{k}}}.
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