# Equation Of Parabola When Focus And Directrix Is Given Pdf

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- 8.4: The Parabola
- Equation of a parabola from focus & directrix
- 8.4: The Parabola
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## 8.4: The Parabola

Number and Algebra : Module 35 Year : PDF Version of module. The parabola is a curve that was known and studied in antiquity. It arises from the dissection of an upright cone. With the advent of coordinate geometry, the parabola arose naturally as the graph of a quadratic function.

The parabola also appears in physics as the path described by a ball thrown at an angle to the horizontal ignoring air resistance. The vertex of the parabola gives information regarding maximum height and combined with the symmetry of the curve also tells us how to find the horizontal range. Quadratic functions frequently appears when solving a variety of problems. Their study in Year 10 gives an excellent introduction to important ideas that will be encountered in senior mathematics and beyond.

This graph is called a parabola. The path of a ball tossed under gravity at an angle to horizontal roughly traces out a parabola. This basic parabola can be reflected in vertical and horizontal lines and translated to produce congruent parabolas.

It is also possible to reflect in other lines and rotate parabolas, but the corresponding algebra is difficult and is generally studied in later y ears using matrices. Reflection in the x -axis. Translations of the basic parabola. When we translate the parabola vertically upwards or downwards, the y -value of each point on the basic parabola is increased or decreased. The vertex of this parabola is now 0, 9 , but it has the same axis of symmetry. We can combine the two transformations and shift parabolas up or down and then left or right.

Finding the y -intercept is a useful thing to do and assists in drawing the diagram. The technique of completing the square enables us the change the given equation to our desired form.

To complete the square, we add and subtract the square of half the coefficient of x. Sketch the following parabolas. First find the y -intercept, then complete the square to find the axis of symmetr y and the vertex of the parabola, then find the x -intercepts if the y exist.

We have seen in the examples so far that some parabolas cut the x -axis and some do not. This will produce a quadratic equation. As discussed in the module, Quadratic equations , this can be solved in three ways:.

Completing the square, which is often done to find the vertex and axis of symmetry anyway, is often the most efficient way of laying bare all of the features of the parabola. This method will, of course, work even if the x -intercepts are surds.

Since the right-hand side is always at least 2, the y -values are never zero. Thus this parabola has no x -intercepts. The problem of completing the square for equations of upside-down parabolas is tricky.

We can then treat the quadratic in the brackets in the usual way. To find the x -intercepts, we set and so. In each case complete the square and determine the x- and y -intercepts, the axis of symmetr y and the vertex of the parabola. There is a further transformation that results in stretching the arms of the parabola, producing a new parabola that is not congruent to the original one. Completing the square for non-monic quadratics.

The following material should be regarded as extension, since it is tricky and the use of calculus in the senior syllabus can also be used to find the vertex. This is demonstrated in the following example. Find the y -intercept, the axis of symmetr y and the vertex of the parabolas b y completing the square. Sketch their graphs. The axis of symmetry is a useful line to find since it gives the x -coordinate of the vertex.

This gives us the equation of the axis of symmetry and also the x -coordinate and the y -coordinate of the vertex. Find its equation. Symmetry and the x -intercepts. We have seen that the parabola has an axis of symmetry. In the case when the parabola cuts the x -axis, the x -coordinate of the axis of symmetry lies midway between the two x -intercepts. Hence the x -coordinate of the vertex is the average of the x -intercepts. We can use this property in some instances to sketch the parabola. Factor if necessary, and sketch, marking intercepts, axis of symmetr y and vertex.

We have emphasized completing the square because it is a such a useful technique and quickly reveals most of the important features of the parabola. If we are given a parabola in factored form, then we can sketch it without expanding and completing the square. Applications involving quadratics.

For example, in physics, the displacement of a particle at time with initial velocity and acceleration is given by. Thus, given the values of u and a , the graph of s against t is a parabola. What is the maximum height that the ball will reach? This is the time at which the ball reaches its maximum height. The example above is one of a host of problems where we try to find the value of one variable that will minimise or maximise another. In senior mathematics a more powerful technique using differential calculus will be used to achieve this.

A farmer needs to construct a small rectangular paddock using a long wall for one side of the paddock. He has enough posts and wire to erect m of fence.

What are the dimensions of the paddock if the fences are to enclose the largest possible area? Let x m be the length of the side perpendicular to the wall. Let A m 2 be the area of the paddock. Thus the dimensions of the paddock are 50m b y m, and the area is square metres. What is the maximum possible area of the rectangle? What are the coordinates of the vertices of such a rectangle? There is a nice graphical approach to solving this problem. Thus we can draw the graph as shown.

Quadratics are sometimes called equations of degree 2. Equations of general degree are called polynomials and are covered in detail in the module Polynomials. Note that this is equal to the discriminant of the quadratic, so that if the roots are equal, the discriminant is 0. Newton developed the theory of symmetric functions and introduced the so called Newton identities that arise in higher algebra. These apply to the roots of polynomials. While the quadratic equation and the parabola were known from the days of the Greeks, higher order curves were not studied in depth until the calculus was developed.

Its graph is shown below. Focus-directrix definition of a parabola. There is also a geometric definition of the parabola in terms of the path traced out by a moving point. Such a graph is known as a locus.

Suppose a is a positive real number. Now find all the points in the plane whose distance from S is equal to its perpendicular distance from the line d. Clearly the origin is one of those points. The point S is called the focus of the parabola and the line d is called the directrix. The real number a is called the focal length of the parabola. The physical applications of this are discussed in the following section.

Some of the history of quadratic equations was covered in the module Quadratic Equations. The Parabola originated in Greek geometry from the dissection of a right-cone. Other curves such as the ellipse, the circle and the hyperbola are obtained by intersecting the cone with other planes. These are shown in the following diagram. We mentioned above that the parabola can also be viewed as the path traced out by a point moving so that its distance from a fixed point, the focus , is equal to its distance from a fixed line called the directrix.

The parabola, viewed in this way, has the remarkable reflection property that a beam of light or radio wave coming into the parabola along a line parallel to the axis of symmetry will reflect off the parabola and pass through the focus. This enables radio waves, light etc to be collected at one point. Conversely, a light source or radio wave emanating from the focus and reflecting off the parabola will move in a direction parallel to the axis of symmetry of the parabola.

This property has made the parabola the standard shape used for satellite dishes, reflecting telescopes as well as spotlights and car headlights. Parabolic reflector. All rays entering parallel to the axis reflect to the focus. The parabola was then studied algebraically as well as geometrically.

Armed with these new tools great minds such as Newton and Leibniz were able to develop the calculus and the parabola and higher degree curves were able to be studied in a new way. Carlyle found an interesting way to solve a quadratic equation graphically using circles and lines. The following exercise exhibits the method.

## Equation of a parabola from focus & directrix

In mathematics , a parabola is a plane curve which is mirror-symmetrical and is approximately U- shaped. It fits several 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 section , created 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 ".

Here we have discussed the steps required for graphing a parabola. Graph the following parabola. Try for free. Write the equation of the circle in standard form given the endpoints of the diameter: , 10 and , Simplify any fractions. Below is a computation using.

Focus and Directrix of a Parabola. Focus: fixed point and directrix are equidistant from the vertex that are the same distance from a fixed point F as they are from a fixed line D. Graph, identify the 6 characteristics, and write the equation.

## 8.4: The Parabola

In The Ellipse we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola. Like the ellipse and hyperbola , the parabola can also be defined by a set of points in the coordinate plane.

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With just two of the parabola's points, its vertex and one other, you can find a parabolic equation's vertex and standard forms and write the parabola algebraically. Just as a quadratic equation can map a parabola, the parabola's points can help write a corresponding quadratic equation. Find an equation describing the door given that is 4 feet across and 8 feet high in the center. The equation of a horizontal ellipse in standard form is where the center has coordinates the major axis has length 2 a, the minor axis has length 2 b , and the. TEKS addressed: b Knowledge and skills. Identify the relationship between a parabola and its focus and directrix, Identify examples of some real world parabolas, Identify the function of some real world parabolas Identify how the focus of a parabola is used in its real world function, Identify if a curve is a parabola.

Number and Algebra : Module 35 Year : PDF Version of module. The parabola is a curve that was known and studied in antiquity. It arises from the dissection of an upright cone. With the advent of coordinate geometry, the parabola arose naturally as the graph of a quadratic function.

Did you know that the Olympic torch is lit several months before the start of the games? The ceremonial method for lighting the flame is the same as in ancient times. The ceremony takes place at the Temple of Hera in Olympia, Greece, and is rooted in Greek mythology, paying tribute to Prometheus, who stole fire from Zeus to give to all humans. Parabolic mirrors or reflectors are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance.

#### Writing Equations of Parabolas in Standard Form

These standard forms are given below, along with their general graphs and key features. Identify and label the vertex , axis of symmetry , focus , directrix , and endpoints of the focal diameter. Thus, the axis of symmetry is parallel to the x -axis. It follows that:. Next we plot the vertex, axis of symmetry, focus, directrix, and focal diameter, and draw a smooth curve to form the parabola. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter.

Conics Worksheet 4 Parabolas Answers 3 axes plt. The Workbook with answers provides opportunities for further practice of new language and exam skills either at home or in the classroom. A paid subscription to ESL Library. Play this game to review Pre-calculus. Some of the worksheets displayed are Volume of solids with known cross sections, 12 cross sections, Solids nets and cross sections, Calculus work on volume by cross sections, Ma work 11 volumes of solids with known cross, Cross section lesson, Cross sections practice, Work solids with known cross sections. Th e four conic sections you have created are known as non-degenerate conic sections.

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Write an equation to represent the cross section of the parabolic plate shown. SOLUTION: The vertex is (0, 0) and p = The parabola opens upward, so is.

The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0,0), then the equation takes one of the following forms. x2.

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