The parabola Presenter Name : Kamalasai School, Kalasin Province Mr.Pramote Phothisai (a,0) (-a,0) v

Objective 1.Determine the equation of the parabola when given a vertex and a focus. 2. Determine the coordinates of the vertex and the focus of the parabola described by the equation.

Interesting Words Entirely Parabola Focus Vertex Curve Upward Downward Equidistant Directrix Length

Phrases referring to what we will talk about A parabola is the locus of all points In a plane equidistant from a fixed point Between the focus and the directrix Is called the vertex of the parabola The curve lies entirely The curve opens upward The curve opens downward

The parabola A parabola is the locus of all points in a plane Point V which lies halfway between the focus and the directrix, is called the vertex of the parabola. v (a,0) (-a,0) directrix X Y equidistant from a fixed point, called the focus, and a fixed line, called the directrix.

The distance from the point (x,y) on the curve to the focus (a,0). (a,0)(-a,0) v X Y (x,y) directrix The distance from the point (x,y) on the directrix x = -a. Since by definition these two distance are equal, we may set them equal : We have an equation for the parabola: The curve must be entirely to the right of the Y-axis.

(a,0)(-a,0) v X Y For example F(2,0) F(3,0) Directrix x = -2 Directrix x = -3 Directrix V (0,0)

If the equation is : The curve lies entirely to the left of the Y-axis. (-a,0)(a,0) v For example F(-2,0) F(-3,0) Directrix x = 2 Directrix x = 3 Directrix V (0,0)

Find the equation of a parabola given the coordinates of its focus and vertex. Standard form Example 1 1) Focus (5, 0) and vertex at origin 2) Focus (-4, 0) and vertex at origin Solution1) Step 1 Step 2 5 Step 3 2) Step 1 Standard form Step 2 4( )x -4 Step 3

If the form of equation is : The curve opens upward and the focus is a point on the Y-axis. v (0,a) (0,-a) For example V (0,0) F(0,2) F(0,3) Directrix y = -2 Directrix y = -3 Directrix

If the form of equation is : The curve opens downward and the focus is a point on the Y-axis. v (0,a) (0,-a) V (0,0) For example F(0,-2) F(0,-3) Directrix y = 2 Directrix y = 3 Directrix

Find the equation of a parabola given the coordinates of its focus and vertex. Standard form Example 2 1) Focus (0, 7) and vertex at origin 2) Focus (0, -5) and vertex at origin Solution1) Step 1 Step 2 7 Step 3 2) Step 1 Standard form Step 2 4( )y -5 Step 3 4( )y

Kamalasai School, Kalasin Province up to now, all of the parabolas we have dealt with have had a vertex at the origin and a corresponding equation in one of the four following forms : 1) 2) 3) 4)

We will now present four more forms of the equation of a parabola. Each one is a standardize parabola with its vertex at point V(h, k). When the vertex is move from the When the vertex is move from the origin to the point V(h, k), the x and y terms of the equation are replaced by (x-h) and (y-k). Then the standard equation for the parabola that opens to the right is :

Example 3Reduce the equation Solution Rearrange the equation so that the second-degree term and any first-degree terms of the same unknown are on the left side. Then group the unknown term appearing only in the first- degree and all constants on the right : Then complete the square in y:

To get the equation in the form Factor an 8 out of the right side. Is the equation of the parabola with its vertex at (-1,3)

Example 4 Solution Reduce the equation Is the equation of the parabola with its vertex at (-1,-4)

OK. That’s it for today. See you next time. Thank you for your attention