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In this section, you will:
• Recognize characteristics of parabolas.
• Understand how the graph of a parabola is related to its quadratic function.
• Determine a quadratic function’s minimum or maximum value.
• Solve problems involving a quadratic function’s minimum or maximum value.

Curved antennas, such as the ones shown in [link] , are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.

In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.

## Recognizing characteristics of parabolas

The graph of a quadratic function is a U-shaped curve called a parabola . One important feature of the graph is that it has an extreme point, called the vertex    . If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value . In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry    . These features are illustrated in [link] .

The y -intercept is the point at which the parabola crosses the y -axis. The x -intercepts are the points at which the parabola crosses the x -axis. If they exist, the x -intercepts represent the zeros     , or roots , of the quadratic function, the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ at which $\text{\hspace{0.17em}}y=0.$

## Identifying the characteristics of a parabola

Determine the vertex, axis of symmetry, zeros, and $\text{\hspace{0.17em}}y\text{-}$ intercept of the parabola shown in [link] .

The vertex is the turning point of the graph. We can see that the vertex is at $\text{\hspace{0.17em}}\left(3,1\right).\text{\hspace{0.17em}}$ Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is $\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ This parabola does not cross the $\text{\hspace{0.17em}}x\text{-}$ axis, so it has no zeros. It crosses the $\text{\hspace{0.17em}}y\text{-}$ axis at $\text{\hspace{0.17em}}\left(0,7\right)\text{\hspace{0.17em}}$ so this is the y -intercept.

## Understanding how the graphs of parabolas are related to their quadratic functions

The general form of a quadratic function presents the function in the form

$f\left(x\right)=a{x}^{2}+bx+c$

where $\text{\hspace{0.17em}}a,b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ are real numbers and $\text{\hspace{0.17em}}a\ne 0.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}a>0,\text{\hspace{0.17em}}$ the parabola opens upward. If $\text{\hspace{0.17em}}a<0,\text{\hspace{0.17em}}$ the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.

The axis of symmetry is defined by $\text{\hspace{0.17em}}x=-\frac{b}{2a}.\text{\hspace{0.17em}}$ If we use the quadratic formula, $\text{\hspace{0.17em}}x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a},\text{\hspace{0.17em}}$ to solve $\text{\hspace{0.17em}}a{x}^{2}+bx+c=0\text{\hspace{0.17em}}$ for the $\text{\hspace{0.17em}}x\text{-}$ intercepts, or zeros, we find the value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ halfway between them is always $\text{\hspace{0.17em}}x=-\frac{b}{2a},\text{\hspace{0.17em}}$ the equation for the axis of symmetry.

[link] represents the graph of the quadratic function written in general form as $\text{\hspace{0.17em}}y={x}^{2}+4x+3.\text{\hspace{0.17em}}$ In this form, $\text{\hspace{0.17em}}a=1,b=4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}c=3.\text{\hspace{0.17em}}$ Because $\text{\hspace{0.17em}}a>0,\text{\hspace{0.17em}}$ the parabola opens upward. The axis of symmetry is $\text{\hspace{0.17em}}x=-\frac{4}{2\left(1\right)}=-2.\text{\hspace{0.17em}}$ This also makes sense because we can see from the graph that the vertical line $\text{\hspace{0.17em}}x=-2\text{\hspace{0.17em}}$ divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, $\text{\hspace{0.17em}}\left(-2,-1\right).\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}x\text{-}$ intercepts, those points where the parabola crosses the $\text{\hspace{0.17em}}x\text{-}$ axis, occur at $\text{\hspace{0.17em}}\left(-3,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-1,0\right).$

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
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?