POWERPOINT ON PRE-CALCULUS FOR DEMO.pptx

This is a slide show on parabola

1. PARABOLAS

2. OBJECTIVES: 1. define a parabola; 2. determine the standard form of equation of a parabola; and 3. graph a parabola in a rectangular coordinate syste m.  

3. PARABOLA - is the set of all points P, such that its distances from a given point F and a given line , are the same. The point F is called the focus of the parabola, and is the directrix of the parabola. The point V , the point on the parabola that is closest to the directrix , is the vertex.  

4. (1) vertex : origin V (0, 0) • If the parabola opens upward, the vertex is the lowest point. If the parabola opens downward, the vertex is the highest point.   ( 2) directrix : the line y = −c or y = c • The directrix is c units below or above the vertex .   ( 3) focus : F(0, c) or F(0,−c) • The focus is c units above or below the vertex. Any point on the parabola has the same distance from the focus as it has from the directrix .   ( 4) axis of symmetry: x = 0 (the y-axis) • This line divides the parabola into two parts which are mirror images of each other.

5. Example : Determine the focus and directrix of the parabola with the given equation . Sketch the graph, and indicate the focus, directrix , vertex, and axis of symmetry.   x 2 = 12y

6. SOLUTION: Vertex (0,0) c = 3 Focus (0 , 3 ) The directrix : y = − 3 axis of symmetry is x = 0.  

7. The following observations are worth noting. The equations are in terms of x − h and y − k: the vertex coordinates are subtracted from the corresponding variable. Thus, replacing both h and k with 0 would yield the case where the vertex is the origin. For instance, this replacement applied to (x − h) 2 = 4c(y − k) (parabola opening upward) would yield x 2 = 4cy, the first standard equation we encountered (parabola opening upward, vertex at the origin ). If the x-part is squared, the parabola is “vertical”; if the y-part is squared, the parabola is “horizontal.” In a horizontal parabola, the focus is on the left or right of the vertex, and the directrix is vertical. If the coefficient of the linear (non-squared) part is positive, the parabola opens upward or to the right; if negative, downward or to the left.

8. Example: The figure shows the graph of parabola , with only its focus and vertex indicated. Find its standard equation . What is its directrix and its axis of symmetry ?

9. Solution : The vertex is V (5, − 4) and the focus is F(3, − 4). From these, we deduce the following: h = 5, k = − 4, c = 2 (the distance of the focus from the vertex). Since the parabola opens to the left, we use the template (y − k) 2 = − 4c(x − h). Our equation is (y + 4) 2 = − 8(x − 5 ). Its directrix is c = 2 units to the right of V, which is x = 7. Its axis is the horizontal line through V : y = − 4 .

10. Directions: Determine the vertex, focus, directrix , and axis of symmetry of the parabola with equation Sketch the graph , and include these points and lines .