In conic sections, a parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix, and the fixed point (F) is called the parabola’s focus. A line through the focus and perpendicular to the directrix is called the axis of the parabola. Also, the vertex is the point of intersection of a parabola with the axis. As we know, the standard equation of the parabola is used in solving a variety of problems in maths. In this article, you will learn how to write the standard equations for parabola in different cases and how to solve questions based on these equations. Show
Standard Equation for ParabolaIn this section, one can learn what is the standard equation of a parabola and how to write the equations of parabola. The parabola equation is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis and y-axis. The four such possible orientations of the parabola are explained in the table below:
Thus, we can derive the equations of the parabolas as: y2 = 4ax y2 = -4ax x2 = 4ay x2 = -4ay These four equations are called standard equations of parabolas. It is important to note that the standard equations of parabolas focus on one of the coordinate axes, the vertex at the origin. The directrix is parallel to the other coordinate axis. However, we can list the following observations from the above equations of parabolas in standard form.
Watch The Below Video To Understand the Orientations of Parabola
 Now, let us write the equation of parabola in standard form when the vertex is not at the origin and the axis of symmetry is parallel to the x-axis or y-axis. These are tabulated as given below:
Solved ExamplesExample 1: Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola y2 = 12x. Solution: Given equation of the parabola is: y2 = 12x Comparing with the standard form y2 = 4ax, 4a = 12 a = 3 The coefficient of x is positive so the parabola opens to the right. Also, the axis of symmetry is along the positive x-axis. Therefore, Focus of the parabola is (a, 0) = (3, 0). Equation of the directrix is x = -a, i.e. x = -3 or x + 3 = 0. Length of the latus rectum = 4a = 4(3) = 12 Example 2: Find the equation of the parabola which is symmetric about the y-axis, and passes through the point (3, -4). Solution: Given that the parabola is symmetric about the y-axis and has its vertex at the origin. Thus, the equation can be of the form x2 = 4ay or x2 = -4ay, where the sign depends on whether the parabola opens upwards or downwards. Since the parabola passes through (3, -4) which lies in the fourth quadrant, it must open downwards. So, the equation will be: x2 = -4ay Substituting (3, -4) in the above equation, (3)2 = -4a(-4) 9 = 16a a = 9/16 Hence, the equation of the parabola is: x2 = -4(9/16)y Or 4x2 = -9y Go through the practice questions given below to get a thorough understanding of the different cases of parabolas explained above. Practice Problems1. Find the equation of the parabola with focus (4, 0) and directrix x = -4. 2. Find the equation of the parabola whose vertex is (0, 0), passing through (5, 2) and symmetric with respect to y-axis. 3. Given the equation of parabola: (x – 2)2 = -8(y – 3) State whether the parabola opens upward, downward, right or left, and also write the coordinates of the vertex, the focus, and the equation of the directrix. 4. Find the focus and directrix of the parabola whose equation is y2 – 6y + 3x + 18 = 0.
The standard form of a parabola that opens left or right, or a horizontal parabola, is as follows: x = a(y - k)2 + h. View complete answer on math.libretexts.org What is the standard equation of the parabola when it opens to the right?use the value of k to determine the axis of symmetry, y=k. set 4p equal to the coefficient of (x−h) in the given equation to solve for p. If p>0,the parabola opens right.View complete answer on math.libretexts.org What equation moves a parabola left and right?The graph of y=(x-k)²+h is the resulting of shifting (or translating) the graph of y=x², k units to the right and h units up. For example, y=(x-3)²-4 is the result of shifting y=x² 3 units to the right and -4 units up, which is the same as 4 units down.View complete answer on khanacademy.org What makes a function move left or right?Moving left and rightThis is always true: To shift a function left, add inside the function's argument: f (x + b) gives f (x)shifted b units to the left. Shifting to the right works the same way; f (x − b) is f (x) shiftedb units to the right. View complete answer on purplemath.com What is the standard form of a parabola?If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y - k)2 = 4p(x - h), where p≠ 0. The vertex of this parabola is at (h, k). The focus is at (h + p, k).View complete answer on sparknotes.com Finding the standard form of a parabola given vertex and focusHow do you write a parabola in standard form?For parabolas that open either up or down, the standard form equation is (x - h)^2 = 4p(y - k). For parabolas that open sideways, the standard form equation is (y - k)^2 = 4p(x - h). The vertex or tip of our parabola is given by the point (h, k).View complete answer on study.com Which parabola opens to the right?If a is positive, the parabola opens up or to the right. If it is negative, it opens down or to the left. The vertex is at (h, k).View complete answer on softschools.com What is the standard equation of a parabola that opens right and whose vertex is at the origin?The standard equation of a parabola with vertex at the origin and vertical orientation is 4py = x2, where p is the distance between the vertex and the origin.View complete answer on academic.sun.ac.za What is the equation of parabola?The simplest equation of a parabola is y2 = x when the directrix is parallel to the y-axis. In general, if the directrix is parallel to the y-axis in the standard equation of a parabola is given as: y2 = 4ax.View complete answer on byjus.com What is the opening of the parabola?In the graph, the highest or lowest point of a parabola is the vertex. The vertex of the graph of y=x2 is (0,0). If a>0 in f(x)=ax2+bx+c, the parabola opens upward. In this case the vertex is the minimum, or lowest point, of the parabola.View complete answer on varsitytutors.com How do you find the standard form?The standard form of a linear equation is Ax+By=C. To change an equation written in slope-intercept form (y=mx+b) to standard form, you must get the x and y on the same side of the equal sign and the constant on the other side.View complete answer on brightstorm.com How do I write an equation in standard form?The standard form for linear equations in two variables is Ax+By=C. For example, 2x+3y=5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.View complete answer on khanacademy.org How do you write a quadratic equation in standard form?The standard form of quadratic equation is ax2 + bx + c = 0, where 'a' is the leading coefficient and it is a non-zero real number. This equation is called 'quadratic' as its degree is 2 because 'quad' means 'square'.View complete answer on cuemath.com What is B in standard form parabola?The “b” value translates the parabola horizontally across the x-axis. If the “b” value is positive, the parabola moves to the left, and if it's negative the “b” value moves to the right.View complete answer on quadratictoolbox.weebly.com How do you move a quadratic function left and right?Shift left and right by changing the value of hYou can represent a horizontal (left, right) shift of the graph of f(x)=x2 f ( x ) = x 2 by adding or subtracting a constant, h , to the variable x , before squaring. If h>0 , the graph shifts toward the right and if h<0 , the graph shifts to the left. View complete answer on courses.lumenlearning.com How do you shift an equation to the left?In function notation, to shift a function left, add inside the function's argument: f(x + b) shifts f(x) b units to the left. Shifting to the right works the same way, f(x - b) shifts f(x) b units to the right.View complete answer on gladwin.glk12.org Which letter determines if a parabola opens up or down?Parabolas have a shape that resembles (but is not the same as) the letter U. Parabolas may open upward or downward. If the sign of the leading coefficient, a, is positive (a > 0), the parabola opens upward. If the sign of the leading coefficient, a, is negative (a < 0), the parabola opens downward.View complete answer on mathbitsnotebook.com What are the different forms of the equations of parabolas?There are three types of parabolas. The three forms are: vertex form, standard form and intercept form. Each form provides you a different key feature for the graph.View complete answer on study.com |
What is the standard equation of a parabola that opens right and whose vertex is at the origin?
Postagens relacionadas
direito autoral © 2024 cemle Inc.
