vertex

Examples of vertex in the following topics:

• Graphing Quadratic Equations in Vertex Form

  • The vertex form of a quadratic function lets its vertex be found easily.
  • Another common form is called vertex form, because when a quadratic is written in this form, it is very easy to tell where its vertex is located.
  • The vertex form is given by:
  • It is more difficult to convert from standard form to vertex form.
  • Now the expression in the parentheses is a square; we can write $y=(x+2)^2+2.$ Our equation is now in vertex form and we can see that the vertex is $(-2,2).$

• Parts of a Parabola

  • One important feature of the parabola is that it has an extreme point, called the vertex.
  • If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value.
  • In either case, the vertex is a turning point on the graph.
  • The axis of symmetry is a vertical line drawn through the vertex.
  • Notice that, for parabolas with two $x$-intercepts, the vertex always falls between the roots.

• Parabolas As Conic Sections

  • The vertex is the point where the plane intersects the exterior surface of the cone.
  • The vertex is therefore also a point on the cone, and the distance between that point and the cone's central axis is the radius of a circle.
  • To locate the $x$-coordinate of the vertex, cast the equation for $y$ in terms of $a x^2 + b x + c$.
  • The vertex will be at the point:
  • For example, in the parabola $y=x^2$, $a=1$, $b=0$, $c=0$, and the vertex is at $x=0$.

• What is a Quadratic Function?

  • With a quadratic function, pairs of unique independent variables will produce the same dependent variable, with only one exception (the vertex) for a given quadratic function.
  • where $h$ and $k$ are respectively the coordinates of the vertex, the point at which the function reaches either its maximum (if $a$ is negative) or minimum (if $a$ is positive).

• Graphing Quadratic Equations In Standard Form

  • The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex.
  • The coefficients $b$ and $a$ together control the axis of symmetry of the parabola and the $x$-coordinate of the vertex.

• Financial Applications of Quadratic Functions

  • Maximum profit is $5500 (the vertex), which is achieved at $250$ sales.

• What Are Conic Sections?

  • The point halfway between the focus and the directrix is called the vertex of the parabola.
  • The vertex lies at the midpoint between the directrix and the focus.

• Parts of a Hyperbola

  • Finally, draw the curve of the hyperbola by following the asymptote inwards, curving in to touch the vertex on the rectangle, and then following the other asymptote out.

• Standard Equations of Hyperbolas

  • The distance b (not shown in below) is the length of the perpendicular segment from either vertex to the asymptotes.

• Scientific Applications of Quadratic Functions

  • which you should recognize as the vertex form of a quadratic equation.