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Finding the Maximum or Minimum of a Quadratic - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Quadratic graphs have several properties such as shape, vertex and direction that help us solve several types of application questions. The vertex, the quadratic minimum and the quadratic maximum are also helpful when answering problems associated with area, speed and direction. We find the minimum if the parabola opens "up" and the maximum if the parabola opens "down."

So, hopefully by now, you are comfortable with the fact that ax squared plus bx plus c is the graph of a parabola. OK, the A determines what the graph is facing, up or down. And then using different methods we were able to find out exactly what that vertex is.
OK, what we are going to talk about now is and application of this. OK? And we know that a parabola is going to basically look like a U. Whether it is facing upwards or downwards can depend.
What we can do is we can get equations that are an application. So, they're representing an area, speed, or something of that nature. And if we ever want to find a maxim or minimum for that application, all we have to do is look for the vertex. OK?
So, if we want to find a minimum of something, in this graph, it's represented by an upward facing parabola, we just find the vertex and that will be the minimum value. Similarly, if we're looking for the maximum, seeing a ball being tossed up or something like that, the maximum is going to occur at the top of the parabola or the vertex as long as the parabola is facing downwards. OK?
So, the vertex is not only just a spot where the graph turned from increasing to decreasing or vice versa, it also can be the maximum or minimum for an application.

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