parabola ppt for understanding the basics
- 2. Graphing a Parabola In this chapter, we will study equations (functions) defined by quadratic (second-degree) polynomials of the form. These functions are quadratic functions and their graphs are parabolas. the “simplest” parabola to graph is y = f(x) = x2
- 3. Graphing a Parabola A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of the parabola and the fixed point F is called the focus. A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola
- 5. Graphing a Parabola From the standard equations of the parabolas, figures above, we have the following observations: 1. Parabola is symmetric with respect to the axis of the parabola.If the equation has a y 2 term, then the axis of symmetry is along the x-axis and if the equation has an x 2 term, then the axis of symmetry is along the y-axis. 2. When the axis of symmetry is along the x-axis the parabola opens to the (a) right if the coefficient of x is positive, (b) left if the coefficient of x is negative. 3. When the axis of symmetry is along the y-axis the parabola opens (c) upwards if the coefficient of y is positive. (d) downwards if the coefficient of y is negative.
- 6. Latus rectum Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola (Fig a). To find the Length of the latus rectum of the parabola y2 = 4ax (Fig b). By the definition of the parabola, AF = AC. But AC = FM = 2a Hence AF = 2a. And since the parabola is symmetric with respect to x-axis AF = FB and so AB = Length of the latus rectum = 4a.
- 7. Graphing a Parabola An important feature of this parabola is its symmetry with respect to the y-axis. If you folded the graph of y = x2 along the y-axis, the two halves of the graph would coincide because the same value of y is obtained for any value of x and its opposite 2x. The points on one side of the axis of symmetry are mirror images of the points on the other side. For instance, x = 2 and x = -2 both give y=4. Because of this symmetry, the y-axis is called the axis of symmetry or the axis of the parabola. The point (0, 0), where the parabola crosses its axis, is called the vertex of the curve.
- 9. Graphing a Parabola of the form y = f(x) = a(x-h)2 +k So far, we have graphed only parabolas of the form y = a x2 + k. What do you think the graph of y = (x-1 )2 looks like? The shape of the graph is identical to that of y =x2 but it is shifted 1 unit to the right. Thus, the vertex is at (1, 0) and the axis of symmetry is as shown in Figure
- 10. Graphing a Parabola of the form y = f(x) = a(x-h)2 +k
- 11. Graphing a Parabola of the form y = f(x) = ax2 +bx+c
- 12. Graphing a Parabola of the form x = f(y) = ay2 +by+c or x = a(y-k)2 +h