Given a quadratic function, find the domain and range.
Identify the domain of any quadratic function as all real numbers.
Determine whether
is positive or negative. If
is positive, the parabola has a minimum. If
is negative, the parabola has a maximum.
Determine the maximum or minimum value of the parabola,
If the parabola has a minimum, the range is given by
or
If the parabola has a maximum, the range is given by
or
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.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature