TonyC said:
The standard form equation for a parabola is y = ax^2 + bx + c, where a is the coefficient of the squared term, b is the coefficient of the linear term, and c is the constant term.
To graph a parabola from its equation, you can plot points by substituting different values for x and solving for y. Another method is to use the vertex form of the equation, y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola. You can then plot the vertex and two other points to draw the parabola.
The coefficient a determines the shape and direction of the parabola. If a is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards. The value of a also determines the steepness of the curve. A larger absolute value of a results in a steeper curve.
The vertex of a parabola can be found by using the formula h = -b/2a and k = c-b^2/4a, where (h,k) is the vertex. The focus can be found by using the formula (h,k+p), where p = 1/4a. The directrix is a horizontal line y = k-p or a vertical line x = h-p, depending on the orientation of the parabola.
The parabola equation is used in various fields, such as physics, engineering, and economics. Some examples include the trajectory of a thrown object, the shape of satellite dishes, and the cost and revenue curves in business. The parabola also appears in architecture and design, such as in the shape of arches and suspension bridges.