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

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

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

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

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

What is the Equation? distance from P(x, y) to A(x,-p) is distance from P(x, y) to F(0, p) is Therefore:

squaring both sides: (y + p)2 = (x2 + (y-p)2 y2 + 2py + p2 = x2 + y2 -2py + p2 4py = x2

Equation of a Parabola p>0 p<0 p>0 p<0

Try This Sketch the parabola, label the directrix & axis of symmetry for the parabola with vertex at (3,2) & focus at (3,4). Write the equation.

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

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

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

Equation of a Parabola p>0 p<0 p>0 p<0