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# Parabola Help

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## Introduction to Parabolas

A conic section is a shape obtained when a cone is sliced. The study of conic sections began over two thousand years ago and we use their properties today. Planets in our solar system move around the sun in elliptical orbits. The cross-section of many reflecting surfaces is in the shape of a parabola. In fact, all of the conic sections have useful reflecting properties. There are three conic sections—parabolas, ellipses (including circles), and hyperbolas.

### Parabolas

In Quadratic Functions Help, we learned how to graph parabolas when their equations were in the form y = a ( xh ) 2 + k or y = ax 2 + bx + c . Now we will learn the formal definition for a parabola and another form for its equation.

DEFINITION: A parabola is the set of all points whose distance to a fixed point and a fixed line are the same.

The fixed point is the focus . The fixed line is the directrix . For example, the focus for the parabola , and the directrix is the horizontal line y = 7. The point (0, 2) is on the parabola. Its distance from the line y = 7 is 5. Its distance from the focus (−3, 6) is also 5.

The new form for a parabola that opens up or down is ( xh ) 2 = 4 p ( yk ). The vertex is still at ( h , k ), but p helps us to find the focus and the equation for the directrix. The focus is the point (h , k + p ), and the directrix is the horizontal line y = kp . The form for the equation for a parabola that opens to the side is ( yk ) 2 = 4 p ( xh ). The focus for a parabola that opens to the right or to the left is the point ( h + p , k ), and the directrix is the vertical line x = hp . This information is summarized in Table 12.1 and in Figures 12.1 and 12.2.

Table 12.1

 ( x − h ) 2 = 4 p ( y − k ) ( y − k ) 2 = 4 p ( x − h ) The vertex is ( h, k ). The vertex is ( h, k ). The parabola opens up if p is positive and down if p is negative. The parabola opens to the right if p is positive and to the left if p is negative. The focus is ( h, k + p ). The focus is ( h + p, k ). The directrix is y = k − p . The directrix is x = h − p . The axis of symmetry is x = h . The axis of symmetry is y = k .

Fig. 12.1

Fig. 12.2

## Matching the Equation of a Parabola to its Graph Examples

In the following examples, we will be asked to match the equation to its graph. The vertex for each parabola will be at (0, 0). We can decide which graph goes to which equation either by finding the focus or the directrix in the equation and finding which graph has this focus or directrix.

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