Sec 6.1 Conic Sections Parabolas Name: What is a parabola? It is geometrically defined by a set of points or locus of points that are equidistant from a point (the focus) and a line (the directri). To see this idea visually try drawing a straight line at the bottom of a piece of paper with a ruler. Then, place a point in the middle just above the line as shown. Net, fold the paper multiple times in various locations so that the line folds on top of the point and make a crease. The creases will outline a parabola. Why does this happen? Parabolas can be found in many places in everyday life. A few eamples are shown below. Can you guess where the focus point should be in the flash light or the satellite dish? Fountains and Projectiles Flash Light Reflectors Algebraically, parabolas are usually defined in two different forms: Standard Form and Verte Form Let s start with the most basic graph of a parabola, Satellite Dishes M. Winking (Section 6-1) p.99
So, where would the focus point and the directri be in the basic equation of y. This is not a trivial task. We know that the directri should be somewhere below the verte and the distance from the verte to the line should be the same as the distance from the verte to the focus point. A (, ) Consider putting an arbitrary point on the y ais above the verte and call it the focus point at (0, p). We will need to figure out what p is to find the location of the focus. Then, we know that the directri is the same distance away from the verte but on the other side of the parabola and it is a line. So, it the directri would need to be the line at y p. y p The parabola is geometrically defined as the set of point equidistant from a point and a line. So, we already know algebraically the parabola is given by the equation y which would suggest every point on the parabola is basically of the form (, ) (eg. (1,1), (, ), (3,9), etc.). We described the point of the focus point as (0,p) and then, if you think about it carefully the point directly below the general point of the parabola shown in the diagram on the directri would have to be the point (, p). Now, we can just use the distance formula to say that AF AB. ( ) + ( y y ) ( ) + ( y y ) A F A F A B A B (0,p) F B (,-p) ( ) ( ) ( ) 0 + p + ( p) We can square both sides: ( ) ( ) ( ) 0 + p + ( p) Clean things up a bit: ( ) 0 ( ) + p + + p Epand (use F.O.I.L. if necessary): + p + p + p + p Cross out items on both sides (subtract from both sides) + So the focus is located at p + p + p + p Move the p to the right: p p + p + p p Solve for p p Which reduces 1 p 1 0, and the directri is located at 1 y M. Winking (Section 6-1) p.100
It turns out doing nearly the same thing for the verte form of the parabola, ( y k ) a ( h) OR ( h) a ( y k ) we can show in a similar manner that the focus is located inside the mouth of the parabola a vertical distance of 1 from the verte whereas the directri is located the same distance away from the verte on the other a side of the parabola from the focus. 1. Sketch a graph the following Parabolas (Label the Verte, Focus, and Directri) a. ( y ) 1( + 3) b. ( y+ 1) 1 ( ) 8 + 3 1 y c. ( y ) ( ) d. ( ) ( ) M. Winking (Section 6-1) p.101
+ 1 y f. 1 e. ( ) ( ) y 8. Find each of the requested components of a parabola. a. Given the verte of a parabola is located at (,) and has a directri of y, find the location of the focus. b. Given the verte of a parabola is located at (,1) and has a focus at ( 1,1), find the equation of the directri. c. Given the directri of a parabola is y 3 and a focus located at (,1), find the location of the verte. Which way does the parabola open? d. Find the verte of the parabola defined by y 6 + M. Winking (Section 6-1) p.10
3. Find each of the requested components of a parabola. a. Given the verte of a parabola is located at (-, -3) and has a directri of y -6, find the location of the focus and the equation of the parabola in standard form. b. Given the focus of a parabola is located at (1.5,1) and has a directri at.5, find the coordinates of the verte and the equation of the parabola in standard form.. c. Find the verte, directri, and focus of the following parabola defined by: y + 1 + d. Find the verte, directri, and focus of the following parabola defined by: y y + 1 M. Winking (Section 6-1) p.103