Module 7: Conics Lesson Notes Part : Parabolas Parabola- The parabola is the net conic section we ll eamine. We talked about parabolas a little bit in our section on quadratics. Here, we eamine them more thoroughl. The official definition of a parabola is the set of all points (, ) that are equidistant from a fied line, the directri, and a fied point, the focus. A segment drawn perpendicularl from the focus to the line will pass through the parabola s verte. That line is called the ais of smmetr. Here s our basic parabola: = : In order to take a closer look at what I m talking about, grab a sheet of graph paper and graph: = We ll come back to that graph in a minute. Up/Down Left/Right- The parabola is slightl different than the other three conics sections we will talk about. This is because in the other three, both the -term and the -term are squared. In a parabola, onl one of those two terms is squared. the -term is squared, ou have a parabola that opens up or down. Up, of course, would be if the -squared term were positive such as the graph above. The parabola would open down if the -squared term were negative such as: = - (We learned this fact in the Translations unit.) the -term is squared, ou have a parabola that opens left or right. Consider the other basic parabola: =
How would ou graph this? Well, ou d have to solve for : Here s how it would look: The parabola opens to the right if the -squared term is positive: Consider this parabola: What does that one look like? Using good deduction, ou probabl can guess it will open to the left. Here s wh: Graphing those two equations gives: Again for these left/right parabolas, we must graph both equations (the positive and negative square root) at once, much like we did for circles. This is because the left/right parabolas are NOT functions, and will not directl graph on our calculator. Standard Equation of a Parabola- (part a) There are actuall two standard equations for a parabola. This is because there are two different tpes of parabolas. To make things simpler to begin, we re going to be looking at the special case where the parabola has a verte on the origin. In part of the notes, we will look at parabolas who s verte is not at the origin. Here are the standard forms of parabolas who s verte is at the origin.
the -term is squared, the parabola has a vertical ais: 4p the -term is squared, the parabola has a horizontal ais: 4p Directri- The directri is a line associated with the parabola. our parabola has a vertical ais (in other words, is the squared term) the directri will be horizontal, and vice versa. ou have a vertical ais, the equation for the directri is: = -p ou have a horizontal ais, the equation for the directri is: = -p Focus- The focus is a point associated with the parabola. The focus is inside the parabola. our parabola has a vertical ais the coordinates of the focus are: (0, p) our parabola has a horizontal ais the coordinates of the focus are: (p, 0) Eample : Let s eamine: Look at standard form: 4p Rewrite our equation in standard form: I ve put the there to help us answer the net question. What is the value for p? Look at our equation compared to standard form: 4 p Mathematicall, we must have: 4p Solving: p 4 *Therefore, our directri is the line: 4 3
*Our focus is the point: (0, ) 4 Graph this line and point on our graph paper from earlier. Now, remember we said that a parabola is the set of points that are equidistant from the focus and the directri. Well now ou have the opportunit to check that. Pick three points on our parabola. For the first point, draw a segment connecting that point to the focus. Now draw another segment from that point to the directri. Remember that when ou draw the segment to the directri, it should be perpendicular to the directri that s the wa ou measure the distance between a point and a line. Compare the two segments ou just drew. Are the the same length? The should be. Do the same for the other two points. Now lets verif that the above parabola was indeed a parabola. First put the equation into the Standard Form of a Conic. ( Now that it is in Standard Form of a conic, let s state the A, B and C. (because there is not a term), and (because there is not a term). Now let s find the value of the discriminant ( ( ou refer to the table below, ou can see that the discriminant is 0, therefore we have verified it to be a parabola. Circle Ellipse Parabola Hperbola Eample : Find the focus and directri of the parabola: Put the parabola in standard form. In other words, we solve for. Once ou do that, compare what we have to the standard form equation. We can see. 4
4 p 4 p p 8 p So the directri is: 8 The focus is: (0, ) 8 Now lets verif that the above parabola was indeed a parabola. First put the equation into the Standard Form of a Conic. ( Now that it is in Standard Form of a conic, let s state the A, B and C. (because there is not a term), and (because there is not a term). Now let s find the value of the discriminant ( ( ou refer to the table below, ou can see that the discriminant is 0, therefore we have verified it to be a parabola. Circle Ellipse Parabola Hperbola 5
Eample 3: Find the focus and directri of the parabola: Put the parabola in standard form. In other words, we solve for. Once ou do that, compare what we have to the standard form equation. We can see. So the directri is: the focus is: ( Eample 4: Find the focus and directri of the parabola: Put the parabola in standard form. In other words, we solve for. Once ou do that, compare what we have to the standard form equation. We can see. So the directri is: ( ) the focus is: ( ) 6
Eample 5: Find the focus and directri of the parabola: Put the parabola in standard form. In other words, we solve for. Once ou do that, compare what we have to the standard form equation. We can see. So the directri is: ( ) the focus is: ( We can work backward as well. we know the focus or the equation for the directri, we can find the equation for the parabola: Eample 6: Write the standard form of the equation of the parabola with the verte at the origin and the given focus, (. Give the equation for the directri as well. the focus is (5, 0), that means that p = 5. According to our Standard Form information, we re looking at a parabola with a horizontal ais. Now we write standard form for a horizontal ais parabola: 4p We know that p = 5: 4 p 4(5) The standard form equation for this parabola is 4(5) p = 5, then the equation for the directri is = -5 7
Eample 7: Write the standard form of the equation of the parabola with the verte at the origin and the given focus, (. Give the equation for the directri as well. the focus is ( that means that. According to our Standard Form information, we re looking at a parabola with a vertical ais. Now we write standard form for a horizontal ais parabola: We know that p = -3: ( The standard form equation for this parabola is 4( 3) p = -3, then the equation for the directri is = 3. 8