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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.
Satellite dishes.
An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)

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] .

Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.

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 x at which y = 0.

Identifying the characteristics of a parabola

Determine the vertex, axis of symmetry, zeros, and y - intercept of the parabola shown in [link] .

Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).

The vertex is the turning point of the graph. We can see that the vertex is at ( 3 , 1 ) . 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 x = 3. This parabola does not cross the x - axis, so it has no zeros. It crosses the y - axis at ( 0 , 7 ) so this is the y -intercept.

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Got questions? Get instant answers now!

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 ( x ) = a x 2 + b x + c

where a , b , and c are real numbers and a 0. If a > 0 , the parabola opens upward. If a < 0 , 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 x = b 2 a . If we use the quadratic formula, x = b ± b 2 4 a c 2 a , to solve a x 2 + b x + c = 0 for the x - intercepts, or zeros, we find the value of x halfway between them is always x = b 2 a , the equation for the axis of symmetry.

[link] represents the graph of the quadratic function written in general form as y = x 2 + 4 x + 3. In this form, a = 1 , b = 4 , and c = 3. Because a > 0 , the parabola opens upward. The axis of symmetry is x = 4 2 ( 1 ) = −2. This also makes sense because we can see from the graph that the vertical line x = −2 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, ( −2 , −1 ) . The x - intercepts, those points where the parabola crosses the x - axis, occur at ( −3 , 0 ) and ( −1 , 0 ) .

Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
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Haidar Reply
Need help with math
Peya
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litshani
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Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
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Practice Key Terms 7

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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