In mathematicsa parabola is a plane curve which is mirror-symmetrical and is approximately U- shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point the focus and a line the directrix.
The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic sectioncreated from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.
The line perpendicular to the directrix and passing through the focus that is, the line that splits the parabola through the middle is called the " axis of symmetry ". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where the parabola is most sharply curved.
The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The " latus rectum " is the chord of the parabola that is parallel to the directrix and passes through the focus.
Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects lightthen light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs.
Conversely, light that originates from a point source at the focus is reflected into a parallel " collimated " beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves.
This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. They are frequently used in physicsengineeringand many other areas.
The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered a way to solve the problem of doubling the cube using parabolas. The solution, however, does not meet the requirements of compass-and-straightedge construction. The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola.
The name "parabola" is due to Apolloniuswho discovered many properties of conic sections.Parabolas This section created by Jack Sarfaty.
The Parabola is defined as "the set of all points P in a plane equidistant from a fixed line and a fixed point in the plane.
A parabola, as shown on the cables of the Golden Gate Bridge belowcan be seen in many different forms. The path that a thrown ball takes or the flow of water from a hose each illustrate the shape of the parabola. Each parabola is, in some form, a graph of a second-degree function and has many properties that are worthy of examination. Let's begin by looking at the standard form for the equation of a parabola. We now need to complete the square for this equation.
I will assume that you have had some instruction on completing the square; but in case you haven't, I will go through one example and leave the rest to the reader. When completing the square, we first have to isolate the Ax 2 term and the By term from the C term. So the first couple of steps will only deal with the first two parts of the trinomial.
Now, take that 4 and place it inside the parenthetical term.
By adding 4 to the inside of the parenthesis, we have done more than just add 4 to the equation. We have now added 4 times the 3 that is sitting in front of the parenthetical term. So, really we are adding 12 to the equation, and we must now offset that on the same side of the equation. We will now offset by subtracting 12 from that 1 we left off to the right hand side.
We have now successfully completed the square. Now we need to get this into more friendly terms. Andoh the wealth of information we can pull from something like this!
We will find the specifics from this type of equation below. Finding the vertex, line of symmetry, and maximum and minimum value for the defined quadratic function. When we have an equation in this form, we can safely say that the 'h' represents the same thing that 'h' represented in the first standard form that we mentioned, as does the 'k'.
The picture below shows this parabola in the first quadrant. Likewise, had the equation read "-4," then the graph would still be pointed upward, but the vertex would have been four units below the x-axis.
The most obvious thing that we can tell, without having to look at the graph, is the origin.
The origin can be found by pairing the h value with the k value, to give the coordinate h, k. The most obvious mistake that can arise from this is by taking the wrong sign of the 'h. To find the line of symmetry of a parabola in this form, we need to remember that we are only dealing with parabolas that are pointed up or down in nature.
With this in mind, the line of symmetry also known as the axis of symmetry is the line that splits the parabola into two separate branches that mirror each other. The number that goes in this blank will be the x-coordinate of the vertex. In order to visualize the line of symmetry, take the picture of the parabola above and draw an imaginary vertical line through the vertex.
We must keep in mind that the equations for vertical and horizontal lines are the reverse of what you expect them to be. In the line of symmetry discussion, we dealt with the x-coordinate of the vertex; and just like clockwork, we need to now examine the y-coordinate. The y-coordinate of the vertex tells us how high or how low the parabola sits. Once we have identified what the y-coordinate is, the last question we have is whether this number represents a maximum or minimum.
We call this number a maximum if the parabola is facing downward the vertex represents the highest point on the parabolaand we can call it a minimum if the parabola is facing upward the vertex represents the lowest point on the parabola. How do we tell if the parabola is pointed upward or downward by just looking at the equation? As long as we have the equation in the form derived from the completing the square step, we look and see if there is a negative sign in front of the parenthetical term.
If there is no negative sign in front, then the parabola faces upward.By Mary Jane Sterling. Rules representing parabolas come in two standard forms to separate the functions opening upward or downward from relations that open sideways. The standard forms tell you what the parabola looks like — its general width or narrowness, in which direction it opens, and where the vertex turning point of the graph is.
Next, complete the square go to Chapter 2 if you need to know more about this technique on the left side of the equation. The vertex of the parabola is 8, —3. The parabola opens upward, because the x term is squared and the multiplier on the right is positive. The 4 a part of the standard form is actually 4 1if you want to show that the a value is 1.
Notice that you have to add 98 to the right, because the 49 that you added to complete the square is multiplied by 2. Now factor —1 from each term on the right, and then divide both sides by The vertex of the parabola is at 1, —7and the parabola opens to the left.
The coefficient on the right. Complete the square on the left by moving the y and 4 to the right side and adding 9 to each side of the equation. Factor and simplify. Complete the square in the parentheses, and add 8 to the right side.
Divide each side by 2. The parabola opens downward, because the value of 4 athe multiplier on the right, is. Move the x and 9 to the right. When you complete the square, you have to add 12 on the right. Divide each side by 3. The parabola opens to the right, because the value of 4 athe multiplier on the right, is positive.The graph of a quadratic function is a parabola, and its parts provide valuable information about the function.
The graph of a quadratic function is a U-shaped curve called a parabola. This shape is shown below. This is shown below. Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane. One important feature of the parabola 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. Parabolas also have an axis of symmetry, which is parallel to the y-axis.
The axis of symmetry is a vertical line drawn through the vertex. The y -intercept is the point at which the parabola crosses the y -axis. There cannot be more than one such point, for the graph of a quadratic function. The x -intercepts are the points at which the parabola crosses the x -axis.
Recall that if the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the function. Describe the solutions to a quadratic equation as the points where the parabola crosses the x-axis.
The roots of a quadratic function can also be found graphically by making observations about its graph. These are two different methods that can be used to reach the same values, and we will now see how they are related.
Consider the quadratic function that is graphed below. Notice that these are the same values that when found when we solved for roots graphically. Solve graphically and algebraically. Therefore, it has no real roots.
We can verify this algebraically. Therefore, there are no real roots for the given quadratic function. We have arrived at the same conclusion that we reached graphically.
Another common form is called vertex form, because when a quadratic is written in this form, it is very easy to tell where its vertex is located.
The vertex form is given by:. The vertex is [latex] h,k.When you kick a soccer ball or shoot an arrow, fire a missile or throw a stone it arcs up into the air and comes down again Get a piece of paper, draw a straight line on it, then make a big dot for the focus not on the line!
Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line. Keep going until you have lots of little dots, then join the little dots and you will have a parabola! Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus.
We also get a parabola when we slice through a cone the slice must be parallel to the side of the cone. If you want to build a parabolic dish where the focus is mm above the surface, what measurements do you need? Try to build one yourself, it could be fun! Just be careful, a reflective surface can concentrate a lot of heat at the focus.
Hide Ads About Ads. Parabola When you kick a soccer ball or shoot an arrow, fire a missile or throw a stone it arcs up into the air and comes down again Except for how the air affects it.
Conic Sections Geometry Index. So the parabola is a conic section a section of a cone.Vertical scaling for the parabola is changed by the coefficient of x 2. The variable a is often used for this coefficient. So, to begin, our starting or reference parabola formula looks like this:. Since a is the coefficient of x 2our equation that includes vertical scaling looks like this:. Here's the graph for this scaling.
What follows is an animation that presents many vertical scalings for our reference parabola. Note that the value for a is shown within parentheses before x 2.
Watch for what happens to the parabola when a changes between positive and negative values. Let's cover some notation. First a bit about parentheses.
In all of the cases shown above understand that the calculation for y proceeds this way:. Don't do any thinking like the following. Do not multiply a times x and then square:. About reflections: Let's for the moment consider two cases. The first point is just as far above the x-axis as the second point is below it. We say the the second point is the reflection of the first point across the x-axis.
A similar reflection across the x-axis would occur for any x-coordinate. Below we will see a graph showing how this all looks when full parabolas are drawn.
How to Graph Parabolas
The graph below has the reference parabola drawn in transparent light grayand it's reflection across the x-axis drawn in black. Several points on the reference parabola have been noted along with their corresponding points on the reflected parabola:. Understand that the sign of a positive or negative controls the reflection of the reference parabola as we think about our transform.
We can imagine that changing the value of a from 1 to -1 reflects the reference parabola over the x-axis. This reflection of the reference parabola correctly represents the graph of the transformed parabola. Now, let's explain the term 'vertical scaling factor'. For a short time set aside the idea of reflection across the x-axis and work only with positive values of a.
How to Put Equations of Parabolas in Standard Form
Imagine our reference parabola:. If we say that it will be vertically scaled by a factor of 2, what we mean is that every point on the reference parabola has been moved up twice as high factor of 2 from the x-axis to become the transformed parabola.
So, if we have our reference parabola:.Here is an animation showing how parallel radio waves are collected by a parabolic antenna. The parallel rays reflect off the antenna and meet at a point the red dot, labelled Fcalled the focus. Click the "See more" button to see more examples. Each time you run it, the dish will become flatter. The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point called the focus and a given line called the directrix.
See some background in Distance from a Point to a Line. Each colored segment has the same length. Don't miss Interactive Parabola Graphswhere you can explore concepts like focus, directrix and vertex. Find the focal length and indicate the focus and the directrix on your graph.
Note: Even though the sides look as though they become straight as x increases, in fact they do not.Ellisse dati gli assi (utilizzando i fuochi)
The sides of a parabola just get steeper and steeper but are never vertical, either. The Gladesville Bridge in Sydney, Australia was the longest single span concrete arched bridge in the world when it was constructed in See Is the Gateway Arch a Parabola?
Parabola with horizontal axis. On the other hand, a function only has one value of y for each value of x. After sketching, we can see that the equation required is in the following form, since we have a horizontal axis:.
This is a similar concept to the case when we shifted the centre of a circle from the origin. So if the axis of a parabola is verticaland the vertex is at hkwe have. If the axis of a parabola is horizontaland the vertex is at hkthe equation becomes. We found above that the equation of the parabola with vertex hk and axis parallel to the y -axis is. Also, don't miss Interactive Parabola Graphswhere you can explore parabolas by moving them around and changing parameters.
A parabolic antenna has a cross-section of width 12 m and depth of 2 m. Where should the receiver be placed for best reception? Parabolic antenna, width 12 m, height 2m. The receiver should be placed at the focus of the parabolic dish for best reception, because the incoming signal will be concentrated at the focus.
Parabola on the cartesian plane. We place the vertex of the parabola at the origin for convenience and use the equation of the parabola to get the focal distance p and hence the required point. So we need to place the receiver 4.
Parabolic antenna showing focus. First we get a set of data points from observing the height of the ball at various times from the graph I've used the bottom of each circle as the data point :. We can use this to find where the ball will be at any time during the motion. You can also use Microsoft Excel to module a parabola.
After you plot the points, right-click on one of the points and choose "Add Trendline". Choose Polynomial, degree 2. In "Options" you can get Excel to display the equation of the parabola on the chart. All of the graphs in this chapter are examples of conic sections. This means we can obtain each shape by slicing a cone at different angles. If we slice a cone parallel to the slant edge of the cone, the resulting shape is a parabola, as shown.
Is the Gateway Arch a Parabola? How to find the equation of a quadratic function from its graph.