advertisement

10.01 Parabolas I n order to find the center, foci, vertex, and directrix of a parabola; we have to be able to put the equation in standard form. The first thing we should do is picture the parabola to determine its direction. Definition: Standard Equation of a Parabola The standard form of the equation of a parabola with vertex at (h,k) is as follows. (x – h)2 = 4p(y – k) p≠0 Vertical axis, directrix: y = k – p (y – k)2 = 4p(x – h) p≠0 Horizontal axis, directrix: x = h – p The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0,0), the equation takes one of the following forms. X2 = 4py Vertical axis Y2 = 4px Horizontal axis So what does this mean? 1. If the equation is squared in x, then it opens up or down depending on p. p>0 then it opens up p<0 then it opens down (in this case it is the y variable that moves up or down and it will be the one changing for the focus and directrix) 2. If the equation is squared in y, then it opens left or right depending on p. p>0 then it opens right p<0 then it opens left (in this case it is the x variable that moves left or right and it will be the one changing for the focus and directrix) Usually you are not given the standard form. So here is an example of how to convert into the standard form. Find the vertex, focus, and directrix of 𝑦 2 − 6𝑦 + 8𝑥 + 25 = 0 Original equation: The equation is squared in the y-term so it opens left or right. 𝑦 2 − 6𝑦 + 8𝑥 + 25 = 0 Keep the y-terms (the terms that are squared) on the left side and move the x-term and the constant to the right side. 𝑦 2 − 6𝑦 = −8𝑥 − 25 I like to create a space for the number we are going to add on both sides. 𝑦 2 − 6𝑦 + ___ = −8𝑥 − 25 + ___ Remember: we can’t change the equation. So, what we add to one side, we have to add to the other side. Now, take half of the coefficient of the y-term (always the middle term of the squared side) and add it on both sides. 𝑦 2 − 6𝑦 + 32 = −8𝑥 − 25 + 32 Now on the left side factor the equation and simplify the right side. (𝑦 − 3)(𝑦 − 3) = −8𝑥 − 25 + 9 Simplify again (𝑦 − 3)2 = −8𝑥 − 16 Factor out the coefficient on the right side (𝑦 − 3)2 = −8(𝑥 + 2) Lastly factor out a 4 from the coefficient of the right side to find p (𝑦 − 3)2 = 4(−2)(𝑥 + 2) Now that our equation is in the form of (y – k)2 = 4p(x – h) we can easily find h, k and p to find the vertex, focus, and directrix. Our formula tells us that h and k are the opposite of what is in the parentheses. So: h = -2 k=3 p = -2 1. Vertex = (h,k) : (-2,3) 2. Focus: From our p we now know that this parabola opens to the left as p<0. Since the focus has to be inside the parabola, it has to be to the left of the vertex. Foci = (-2-2,3) = (-4,3) 3. Directrix: is a line p units away from the vertex. It lies outside of the vertex. Remember: Horizontal axis, directrix: x = h – p Therefore, Directrix: x = -2-(-2) x = 0