Section 6.1: Parabolas Learning Objectives: 1. Graph parabolas in which the vertex is the origin. 2. Find the equation of a parabola. 3. Graph parabolas in which the vertex is not the origin. 4. Solve applied problems involving parabolas. 
1. The Parabola Parabolas that Open Up or Down The graph of y = a(x – h)^{2} + k or y = ax^{2} + bx + c is a parabola that 1. opens up if a > 0 and opens down if a < 0. 2. has vertex (h, k) if the equation is of the form y = a(x – h)^{2}+ k. 3. has a vertex whose coordinate is The ycoordinate is found by evaluating the equation at the xcoordinate of the vertex y = a(x – h)^{2} + k, a > 0 y = a(x – h)^{2} + k, a < 0 

1. The Parabola A parabola is the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix. In other words, a parabola is a set of points 
2. Equations of a Parabola From the distance formula we can obtain the fact that the equation of the parabola whose vertex is at the origin and opens to the right is y^{2} = 4ax.


Graphing a Parabola Example continued: Graph the equation x^{2} = – 16y. 
Finding the Equation of a Parabola Example: Find an equation of the parabola with vertex at (0, 0) and focus at (– 8, 0). 

3. A Parabola Whose Vertex is Not the Origin

A Parabola Whose Vertex is Not the Origin Example continued: Graph the parabola x^{2} + 2x – 8y + 25 = 0.
