Every parabola has a vertex, an axis of symmetry, a maximum OR minimum value, and a y-intercept. Most but not all also have at least one x-intercept. Let’s define these terms. Vertex: The point where the function changes direction.

There is an easy way to tell whether the graph of a quadratic function opens upward or downward: if the leading coefficient is greater than zero, the parabola opens upward, and if the leading coefficient is less than zero, the parabola opens downward.

In graphs of quadratic functions, the sign on the coefficient a affects whether the graph opens up or down. If a<0 , the graph makes a frown (opens down) and if a>0 then the graph makes a smile (opens up).

If a>0 in f(x)=ax2+bx+c, the parabola opens upward. In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of a makes a narrow parabola; a positive value of a which is close to 0 makes the parabola wide. If a<0 in f(x)=ax2+bx+c, the parabola opens downward.

One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value.

Parabolic Equation. The general equation of parabola is y = x² in which x-squared is a parabola. Work up its side it becomes y² = x or mathematically expressed as y = √x. Formula for Equation of a Parabola.

The hyperbola is the set of all points (x,y) such that the difference of the distances from (x,y) to the foci is constant. The standard form of an equation of a hyperbola centered at the origin with vertices (±a,0) ( ± a , 0 ) and co-vertices (0±b) ( 0 ± b ) is x2a2−y2b2=1 x 2 a 2 − y 2 b 2 = 1 .

The hyperbola is not a function because it fails the vertical line test. Regardless of whether the hyperbola is a vertical or horizontal hyperbola…

No, a circle is a two dimensional shape. No. The mathematical formula used to describe a circle is an equation, not one function. For a given set of inputs a function must have at most one output.

All parabolas are vaguely “U” shaped and they will have a highest or lowest point that is called the vertex. Every parabola has an axis of symmetry and, as the graph shows, the graph to either side of the axis of symmetry is a mirror image of the other side.

The standard form of a parabola is (x – h)2 = a(y – k) or (y – k)2 = a(x – h), where (h, k) is the vertex. The methods used here to rewrite the equation of a parabola into its standard form also apply when rewriting equations of circles, ellipses, and hyperbolas.

If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y – k)2 = 4p(x – h), where p≠ 0. The vertex of this parabola is at (h, k). The focus is at (h + p, k). The directrix is the line x = h – p.