11.3 Parabolas Objectives: Define a parabola. Write the equation of a parabola. Identify important characteristics of parabolas. Graph parabolas.

Parabola The set of all points equidistant from a fixed line called the directrix and a fixed point, the focus.

Standard Equation of a Parabola with Vertex on the Origin Focus: (0, p) (p, 0) Directrix: y = –p x = – p

Characteristics of a Parabola Important Facts: The parabola bends toward the focus and away from the directrix The linear term determines the orientation of the parabola and the axis of symmetry—left/right or up/down The sign of p determines which way the parabola opens The distance between the focus and directrix is 2p The distance from the vertex to the focus and the distance from the vertex to the directrix is p The vertex is the midpoint of the line segment joining the focus and the directrix

Example #1 Show that the graph of the equation is a parabola. Draw its graph, and then find and label its focus and directrix.

Example #2 Graph the following parabola using a graphing calculator.

Example #3 Find the focus, directrix, and equation of the parabola that passes through the point , has vertex (0, 0) and has its focus on the y-axis. Sketch the graph of the parabola, and label its focus and directrix.

Example #4 A certain satellite dish has the shape of a parabolic dish (a cross section through the center of the dish is a parabola). It is 45 cm deep at the center and has a diameter of 274 cm. How far from the vertex of the parabolic dish should the receiver be placed in order to “catch” the transmitted signal? The transmitter should be placed on the focus for best reception.

Example #4 It is 45 cm deep at the center and has a diameter of 274 cm. How far from the vertex of the parabolic dish should the receiver be placed? The receiver should be placed up 104 cm from the base. By drawing this out with vertex on the origin, we can figure out a point on the parabola by dividing the diameter by 2.