Conics: The Parabola Objectives: Use the standard and general forms of the equation of a parabola Graph parabolas ©2002 Roy L. Gover (www.mrgover.com)

Applications Trajectory The Parabola: Parabolic Reflectors

Definition A parabola is the locus (set) of all points that are equidistant from a point called the focus and a line called the directrix.

Important Idea They may point up and down or left and right The graphs of parabolas come in different shapes and sizes

Definition The vertex is the lowest, highest, leftmost or rightmost point on the graph

Definition The vertex is half-way between a point called the focus and a line called the directrix. Vertex Focus Directrix Focus Directrix Vertex

Important Idea The focus is a point and is always inside the parabola Directrix Vertex Vertex Directrix

Important Idea The directrix is a line and is always outside the parabola Focus Focus Directrix Vertex Vertex Directrix

Definition The line connecting the focus and vertex and perpendicular to the directrix is the axis of symmetry

Important Idea The distance between the focus and vertex is p units where p is a real number. p

Important Idea The distance between the vertex and directrix is also p units p These distances are always the same.

Important Idea Every point on the parabola is the same distance from the focus and the directrix

Try This Match the letter to the name of the parts of a parabola: A C

Try This Match the letter to the name of the parts of a parabola: What is true about the distance from C to A and the distance from C to B? C B A C A

Try This What appears to be true about the distance from the focus to the points on the parabola opposite the focus? 2p

Definition Standard Equation for the parabola-2 forms: 1. Opens left if p is negative Opens right if p is positive Vertex is at (h,k) P is distance from vertex to focus

Definition Standard Equation for the parabola-2 forms: 2. Opens down if p is negative Opens up if p is positive Vertex is at (h,k) P is distance from vertex to focus

Definition p>0 p<0 p>0 p<0

Example Sketch the parabola, label the vertex, focus, directrix & axis of symmetry for the parabola with vertex at (2,3), p=2, & directrix parallel to the y axis. Write the equation.

Try This Sketch the parabola, label the vertex, focus, directrix & axis of symmetry for the parabola with vertex at (3,2), p=2, & directrix parallel to the x axis. Write the equation.

Solution h k 4p Vertex: (3,2) Focus:(3,4) Directrix:y=0 Axis of Sym:x=3

Example Graph using a graphing calculator:

Example Graph using a graphing calculator:

Example Sketch the graph of the parabola: Hint: write in standard form by completing the square.

Try This Write the standard form of the following parabola:

Try This Find the coordinates of the vertex and the value of p for the parabola. Which way does the parabola open? Vertex: (0,4);p=4;open up

Lesson Close Which of the following are equations of circles and which are equations of parabolas?