Last night, I was helping a friend in his instructional material for his demo. And I was amazed how my brain still manage to remember the lesson.
You can't go through algebra without seeing quadratic functions. The graphs of quadratic functions are parabolas; they tend to look like a smile or a frown. It is of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.Without the term , this function would be a linear function. That is why we specify that a≠0.
Graphing Quadratic Functions
Any real number may be used for x in the formula . So the domain (the set of x-coordinates) for any quadratic function is the set of all real numbers, R. The graph of a quadratic function is a curve called a parabola. The graphs of quadratic functions are parabolas; they tend to look like a smile or a frown, it may open upward or descending and differ in "width" or "steepness", yet they all have a similar fundamental "U" shape..
The picture above shows different graphs, and they are all parabolas.
If a>0 in the equation , then the parabola opens upward. If a<0, then the parabola opens downward.
The Vertex and the Intercepts of a Parabola
The lowest point on a parabola that opens upward or the highest point on the parabola that opens downward is called the vertex. The y-coordinate of the vertex is the minimum value of the function if the parabola opens upward, and it is the maximum value of the function if the parabola opens downward.
Because the vertex is either the highest or lowest point on a parabola, it is an important point to find before drawing the graph. The vertex can be found using the following fact.
Vertex of a Parabola
The x-coordinate of the vertex is -b/2a , provided that a≠0. When you graph a parabola, you should always locate the vertex because it is the point at which the graph “turns around”. With the vertex and several nearby points you can see the correct shape of the parabola.
Last night's overnight was just a proof that whenever you help someone your are also being helped.