Finding the domain and range of a quadratic function
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.
Determining the maximum and minimum values of quadratic functions
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.
Finding the maximum value of a quadratic function
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.
Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length
What dimensions should she make her garden to maximize the enclosed area?
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.
We know we have only 80 feet of fence available, and
or more simply,
This allows us to represent the width,
in terms of
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
The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since
is the coefficient of the squared term,
and
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.
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills