Vertex of a Parabola - Maple Help

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Vertex of a Parabola

Main Concept

The vertex of a parabola is the point at the intersection of the parabola and its line of symmetry.


For a parabola whose equation is given in standard form y&equals;a x2&plus;b x&plus;c, the vertex will be the minimum (lowest point) of the graph if a&gt;0 and the maximum (highest point) of the graph if a<0.


Vertex form

The vertex form of a parabola is y&equals; axh2  &plus;k  where h&comma;k is the vertex. The variable a has the same value and function as the variable in the standard form. If a &gt; 0  (positive)  the parabola opens up, and if  a < 0 (negative) the parabola opens down.


The vertex form of a parabola is usually not provided. To convert from the standard form y&equals;ax2 &plus; bx &plus; c to vertex form, you must complete the square.

Example of completing the square:

1. Factor the leading coefficient out of the first two terms.

4 x2&plus;8 x&plus;4  &equals;

4x2 &plus;2 x &plus;4

2. Complete the square by addition and subtracting the magic number - the square of half the coefficient of x.


4x2&plus; 2 x  &plus; 1  1 &plus;4

3. Factor out the constant b2 a2  .


4x2&plus; 2 x  &plus; 1  &plus;4 4

4. Factor the perfect square.


4x&plus; 1 2


Use the sliders to change the vertex h&comma;k and observe how your changes affect the graph of the parabola.

Observe that when the graph opens up, the range of the corresponding quadratic function is k&comma;, while if it opens down, the range is &comma;k.




Value of a





Value of k:


   Value of h:

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