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Class Note for MATH 1330 at UH 3

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This 12 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 27 views.

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Date Created: 02/06/15
Math 1330 Section 81 Equations of Parabolas A parabola is the set of all points in the plane that are equidistant from a fixed line called the directrix and a fixed point not on the line called the focus The line that passes through the focus and is perpendicular to the directrix is called the axis of the parabola The vertex of the parabola is the point on the parabola that is halfway between the focus and the directrix focal chord 39Zpap 2 FO p focus p p V 0 X V6 ex directrix Basic Equation of a Parabola x24pypgt0 This parabola opens upward Vertex 0 0 Focus 0 p Directrix y p Axis x 0 Focal width 4p Endpoints of focal chord 2p p and 2p p Basic Equation of a Parabola x2 4pypgt0 This parabola opens downward Vertex 0 0 Focus 0 p Directrix y p Axis x 0 Focal width 4p Endpoints of focal chord 2p p and 2p p Basic Equation of a Parabola y24pxpgt0 This parabola opens to the right Vertex 0 0 Focus p 0 Directrix x p Axis y 0 Focal width 4p Endpoints of focal chord p 2p and p 2p Basic Equation of a Parabola y2 4px 19 gt 0 This parabola opens to the left Vertex 0 0 Focus p 0 Directrix x p Axis y 0 Focal width 4p Endpoints of focal chord p 2p and p 2p Translations To graph xh2 4py k 9 gt 0 xh2 4py k 9 gt 0 y k2 4190641 P gt 0 y k2 4Pxh p gt 0 shift the graph of the parabola x2 417 p gt 0 x2 4py p gt 0 y2 4px p gt 0 y2 4px p gt 0 respectively horizontally h units and vertically k units If h gt 0 the horizontal shift is to the right If h lt 0 the horizontal shift is to the left If k gt 0 the vertical shift is upward If k lt 0 the vertical shift is downward The following list re ects the changes in translating the parabola x2 4 py to the parabola 36 h2 4170 k This parabola opens upward Vertex h k Focus hp k Directrix y p k Axis x h Focal width 4p Endpoints of focal chord 2p h p k and 2p h p k The following list re ects the changes in translating the parabola x2 4py to the parabola x h2 4py k This parabola opens downward Vertex h k Focus h p k Directrix y p k Axis x h Focal width 4p Endpoints of focal chord 2p h p k and 2p h p k The following list re ects the changes in translating the parabola y2 41736 to the parabola y 02 417x ll This parabola opens to the right Vertex h k Focus p h k Directrix x p h Axis y k Focal width 4p Endpoints of focal chord p h 2p k and p h 2p k The following list re ects the changes in translating the parabola y2 2 4px to the parabola y k2 4Px This parabola opens to the left Vertex h k Focus 17 h k Directrix x p h Axis y k Focal width 4p Endpoints of focal chord p h 2p k and p h 2p k Example 1 Write the following equation in the standard form for the equation of a parabola x2 6x 12y51 Example 2 Write the following equation in the standard form for the equation of a parabola y28y968x Example 3 Given the parabola x2 16y a Write the given equation in the standard form State the equation of the axis State the coordinates of the vertex State the equation of the directrix State the coordinates of the focus State the focal width State the coordinates of the endpoints of the focal chord Sketch the graph of the parabola which includes the features from c e and g Label the vertex V and focus F P QQWFDFLPFV Example 4 Given the parabola y2 8x 0 a Write the given equation in the standard form State the equation of the axis State the coordinates of the vertex State the equation of the directrix State the coordinates of the focus State the focal width State the coordinates of the endpoints of the focal chord Sketch the graph of the parabola which includes the features from c e and g Label the vertex V and focus F P QQWFDFLPFV Example 5 Given the parabola x 22 4y 1 Write the given equation in the standard form State the equation of the axis State the coordinates of the vertex State the equation of the directrix State the coordinates of the focus State the focal width State the coordinates of the endpoints of the focal chord Sketch the graph of the parabola which includes the features from c e and g Label the vertex V and focus F P QQWFDFLPVEE Example 6 Given the parabola y 12 2 x l a Write the given equation in the standard form State the equation of the axis State the coordinates of the vertex State the equation of the directrix State the coordinates of the focus State the focal width State the coordinates of the endpoints of the focal chord Sketch the graph of the parabola which includes the features from c e and g Label the vertex V and focus F P QQWFDFLPFV Example 7 Given the parabola x2 2x 12y 23 Write the given equation in the standard form State the equation of the axis State the coordinates of the vertex State the equation of the directrix State the coordinates of the focus State the focal width State the coordinates of the endpoints of the focal chord Sketch the graph of the parabola which includes the features from c e and g Label the vertex V and focus F P qorhrvgogrs Example 8 Given the parabola y2 6y 6x 3 0 a Write the given equation in the standard form State the equation of the axis State the coordinates of the vertex State the equation of the directrix State the coordinates of the focus State the focal width State the coordinates of the endpoints of the focal chord Sketch the graph of the parabola which includes the features from c e and g Label the vertex V and focus F P QQWFDFLPFV Tangents t0 Parabolas A line through a point that lies on a parabola is said to be tangent to the parabola at that point provided that the line intersects the parabola only at that point and the line is not parallel to the axis of the parabola It can be shown that a tangent to a point x0 ya that lies on the parabola with equation y ax2 bx c has slope m Zaxo b Example 9 Write an equation of the line tangent to the parabola with equation fx 3x2 6x 1 at x 1 Example 10 Write an equation of the line tangent to the parabola with equation f x 6x2 5x 9 at the point 4 85

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