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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 math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar