# 3.2 Quadratic functions  (Page 4/14)

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Given a quadratic function, find the domain and range.

1. Identify the domain of any quadratic function as all real numbers.
2. Determine whether $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is positive or negative. If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is positive, the parabola has a minimum. If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is negative, the parabola has a maximum.
3. Determine the maximum or minimum value of the parabola, $\text{\hspace{0.17em}}k.$
4. If the parabola has a minimum, the range is given by $\text{\hspace{0.17em}}f\left(x\right)\ge k,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left[k,\infty \right).\text{\hspace{0.17em}}$ If the parabola has a maximum, the range is given by $\text{\hspace{0.17em}}f\left(x\right)\le k,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-\infty ,k\right].$

## Finding the domain and range of a quadratic function

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=-5{x}^{2}+9x-1.$

As with any quadratic function, the domain is all real numbers.

Because $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the $\text{\hspace{0.17em}}x\text{-}$ value of the vertex.

The maximum value is given by $\text{\hspace{0.17em}}f\left(h\right).$

The range is $\text{\hspace{0.17em}}f\left(x\right)\le \frac{61}{20},\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-\infty ,\frac{61}{20}\right].$

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=2{\left(x-\frac{4}{7}\right)}^{2}+\frac{8}{11}.$

The domain is all real numbers. The range is $\text{\hspace{0.17em}}f\left(x\right)\ge \frac{8}{11},\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left[\frac{8}{11},\infty \right).$

## 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.

1. Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length $\text{\hspace{0.17em}}L.$
2. 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, $\text{\hspace{0.17em}}W,\text{\hspace{0.17em}}$ to represent the width of the garden and the length of the fence section parallel to the backyard fence.

1. We know we have only 80 feet of fence available, and $\text{\hspace{0.17em}}L+W+L=80,\text{\hspace{0.17em}}$ or more simply, $\text{\hspace{0.17em}}2L+W=80.\text{\hspace{0.17em}}$ This allows us to represent the width, $\text{\hspace{0.17em}}W,\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}L.$
$W=80-2L$

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 $\text{\hspace{0.17em}}L.\text{\hspace{0.17em}}$ The function, written in general form, is

$A\left(L\right)=-2{L}^{2}+80L.$
2. 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 $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the coefficient of the squared term, $\text{\hspace{0.17em}}a=-2,b=80,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}c=0.$

To find the vertex:

The maximum value of the function is an area of 800 square feet, which occurs when $\text{\hspace{0.17em}}L=20\text{\hspace{0.17em}}$ 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.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this