Question 90016: How are the zeros for a function (the values that make the function value equal zero), and the x-intercepts for the graph of the function related to each other?
A quadratic function has a definite maximum or minimum point (its vertex) that other polynomials may not have. This makes it a good candidate for use in application problems that may require finding a maximum or minimum quantity. If A(x) = -6x2 + 2x + 3 is used to describe a situation involving Area, or Altitude, or Age. How would you go about using the function and its graph to determine the maximum area, altitude or age?
Please explain!!!!
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! 1) The solutions of a quadratic equation are called the "roots" of the equation.
The roots of a quadratic equation can be found by finding the x-intercepts or "zeros" of the related quadratic function. Does this help?
2) You are asking..."how do I find the maximum (or minimum) of a quadratic function?"
Using the given example of  , you can find the maximum or minimum as follows:
First, notice that the coefficient of the  term (-6) is < 0. This means that the parabola represented by the function opens downward so the vertex is at the maximum.
When you are looking for the maximum (or minimum) in such a function, you are really asking..."What is the maximum (or minimum) value of the independent variable?"
The independent variable is easily identified when using the function form because it appears in the parentheses of the function, for example:
Here, the independent variable is x.
To find the x-coordinate of the vertex, which will also be the maximum (or minimum) value of the independent variable, use the following formula:
In your example of , a = -6, b = 2, so...
 This is the x-coordinate of the vertex which, in this case, is the maximum.
So the maximum value of the given function is 1/6.
Let's look at the graph of the function and you should be able to confirm this.
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