A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola, and the line is called the directrix . The directrix is perpendicular to the axis of symmetry of a parabola and does not touch the parabola. If the axis of symmetry of a parabola is vertical, the directrix is a horizontal line .
If we consider only parabolas that open upwards or downwards, then the directrix is a horizontal line of the form y = c . Relation between focus, vertex and directrix:The vertex of the parabola is at equal distance between focus and the directrix. If F is the focus of the parabola, V is the vertex and D is the intersection point of the directrix and the axis of symmetry, then V is the midpoint of the line segment F D ¯ .
Example: If a parabola has a vertical axis of symmetry with vertex at ( 1 , 4 ) and focus at ( 1 , 2 ) , find the equation of the directrix. If F is the focus of the parabola, V is the vertex and D is the intersection point of the directrix and the axis of symmetry, then V is the midpoint of the line segment F D ¯ . Equate the x -coordinates and solve for p . 1 = 1 + p 2 2 = 1 + p p = 1 Equate the y -coordinates and solve for q . 4 = 2 + q 2 8 = 2 + q q = 6 The equation of the directrix is of the form y = c and it passes through the point ( 1 , 6 ) . Here, c = 6 . So, the equation of the directrix is y = 6 . The graph is as shown.
The Parabola
The Northern Cross Radiotelescope http://www.ira.cnr.it/Medicinaweb/Nc/DESCRIPTION.htm
Click here for demonstration of the definition Activity A fun classroom activity involving parabolas is paper folding with wax paper. The idea was obtained from an article by Scott Smith titled Paper Folding and Conic Sections. Instructions On the sheet of wax paper draw a fixed line and a fixed point. Fold the fixed point onto any point on the fixed line and crease the paper. Repeat this many times. The creases form tangents to and envelope a parabola. The following is a GSP simulation of what takes place with the paper folding. This exercise leads to a nice presentation of the derivation of the standard equation of a parabola whose vertex is at the origin. The figure below shows point P called the Focus as the point that is folded onto point P’ (located on The Directrix). At P’ a perpendicular line to the Directrix was constructed. This perpendicular line hits the crease line at point A. The crease line is the perpendicular bisector of PP’, thus point A is equidistant from fixed point A and fixed the fixed line. Point A is a point on the parabola whose focus is P and whose directrix is L. (this can be proved by showing that the crease line through A is tangent to the parabola).
By definition of a parabola the distance from point A to P is equal to the distance from A to P’. Recalling the distance formula
we can write the following the length of = the length of = or by squaring both sides and multiplying out
or the standard form of an up/down parabola whose vertex is at the origin The following activity was obtained from Virginia Laird, Rockwell High School, Richardson, Texas. Students should be provided with the below information through direct teaching and then complete the following problems. • The standard form of a parabola with vertex is and. • A parabola is the set of all points P in a plane that are the same distance from a given line and a fixed point not on the line. • The fixed point is called the focus and the fixed line is called the directrix. • The focus and the directrix are each located c units from the vertex, but lie on opposite sides of the parabola. • The parabola cannot intersect the directrix. • The width of the parabola at its focus is 4c units and is called the latus rectum. • A parabola’s direction is dependent upon which term is squared in the standard form. After the above information has been given to the students they should be grouped in pairs. They should then explore the effects of (h,k) and c upon the parabola in each of the forms. Graph the following parabolas on the computer graphics program and draw conclusions about the direction of the curve, the effect of (h,k) and c. Find the length of the latus rectum for each curve and verify from the graph. Sketch the graph of each on your graph paper. Also multiply each of the equations out and solve for x or y accordingly. 1. 2. 3. 4. Click here for answers Based on the work that you have done above complete the following paragraph to enable you to make some conjectures about parabolas.
For answers click here Describe in paragraph form the following parabolas. Check your results by graphing each parabola on the computer. 1. 2. Graphing Calculator Sketches Put the following equations in standard form. Use above conjectures to construct each parabola on graph paper and use Graphing Calculator to verify graph. 1. 2. Graphing Calculator Sketches Return to main page |