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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

0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply
tan20?×tan40?×tan80?
Santosh Reply
At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?
Kimberly Reply
-3 and -2
Julberte Reply
tan(?cosA)=cot(?sinA) then prove cos(A-?/4)=1/2?2
Chirag Reply
tan(pi.cosA)=cot(?sinA) then prove cos(A-?/4)=1/2?2
Chirag Reply
sin x(1+tan x)+cos x(1+cot x) = sec x +cosec
Ankit Reply
let p(x)xq
Sophie Reply
To the nearest whole number, what was the initial population in the culture?
Cheyenne Reply
do posible if one line is parallel
Fran Reply
The length is one inch more than the width, which is one inch more than the height. The volume is 268.125 cubic inches.
Vamprincess Reply
Using Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating ?T??T? and ?a.?
James Reply
cos(x-45)°=Sin x ;x=?
Samaresh Reply
10-n ft
Nalin Reply
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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