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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 $\text{\hspace{0.17em}}\left(-1,0\right).$

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(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
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Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
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DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
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x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
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The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
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The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
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Soo sorry (5±Root11* i)/3
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Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
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hello
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Miranda Drice
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Miranda
I am living in india
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda