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Find the domain and range of
As with any quadratic function, the domain is all real numbers.
Because is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the value of the vertex.
The maximum value is given by
The range is or
Find the domain and range of
The domain is all real numbers. The range is or
The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the parabola . We can see the maximum and minimum values in [link] .
There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.
A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.
Let’s use a diagram such as [link] to record the given information. It is also helpful to introduce a temporary variable, to represent the width of the garden and the length of the fence section parallel to the backyard fence.
Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so
This formula represents the area of the fence in terms of the variable length The function, written in general form, is
To find the vertex:
The maximum value of the function is an area of 800 square feet, which occurs when feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.
Given an application involving revenue, use a quadratic equation to find the maximum.
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