PreCalc Parabolas Lesson 6 5 We all know

Pre-Calc Parabolas Lesson 6. 5 We all know the general shape of the parabola y = ax 2 Now what if the equation is x = ay 2 ?

Now lets consider the geometric definition of a parabola It is the set of all points P in a plane that are equidistant from a fixed point ‘F’ called the focus point, and a fixed line ‘d’ called the directrix. The ‘focus point’ lies on the axis of symmetry The ‘directrix will be a line ‘perpendicular’ to the axis of symmetry. In general, the distance from the vertex to the focus is ‘p’ units. Also, the distance from the vertex to the directrix is also ‘p’ units.

Now the four different types of parabolas shown earlier can be broken down in to these four basic equations: y = 1 x 2 ; 4 p y = - 1 x 2 ; 4 p x = 1 y 2 4 p ; x = - 1 y 2 4 p (Look at the four basic shapes and their related equations found on page 238) Keep in mind to find the coordinates of the focus, and the equation of the directrix, we need to find the value of ‘p’. To do this we take the coefficient of our squared term and set it equal to + 1 4 p (+) -- if opens up or right; (- ) -- if opens down or left.

Example 1 Find the focus and directrix of each parabola whose equation is given: a. y = 2 x 2 b. x = 1 y 2 20

Example 2: Find an equation of the parabola with vertex (0, 0) and directrix x = 2.

Example 3: Tell whether the parabola x – 1 = - ½ (y + 2) 2 opens up, down, right, or left. Give the coordinates of the: a. vertex b. focus c. The equation of the directrix.

Example 4 Find an equation of the parabola with vertex (0, 0) and focus (0, -3/2). b) Sketch the parabola.