# 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.

can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas