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Theshiftedformofaparabolahomework


How to Master


The Shifted Form of a Parabola


for Your Homework




If you are studying algebra,


you have probably encountered


the concept of a parabola.


A parabola is a curve that has a special shape called a vertex,


which is the highest or lowest point of the curve.


The vertex also determines the axis of symmetry of the parabola,


which is the vertical line that passes through the vertex and divides the parabola into two equal parts.




theshiftedformofaparabolahomework



But what if you want to move the parabola to a different position on the coordinate plane?


How can you change the equation of the parabola to reflect this shift?


This is where


the shifted form of a parabola


comes in handy.


In this article,


we will explain what


the shifted form of a parabola


is,


how to find it from the standard form,


and how to graph it using transformations.


What is


The Shifted Form of a Parabola?




The shifted form of a parabola


is also known as the vertex form or the standard form.


It is an equation that shows how the parabola is shifted horizontally and vertically from its original position.


The shifted form of a parabola


has the following general form:


y = a(x - h) + k


In this equation,


a, h, and k are constants that determine the shape and position of the parabola.


The constant a tells us how wide or narrow the parabola is,


and whether it opens up or down.


If a > 0,


then the parabola opens up and has a minimum vertex.


If a < 0,


then the parabola opens down and has a maximum vertex.


The larger the absolute value of a,


the narrower the parabola.


The smaller the absolute value of a,


the wider the parabola.


The constants h and k tell us how much the parabola is shifted horizontally and vertically from its original position.


The constant h tells us how much the parabola is shifted left or right from the y-axis.


If h > 0,


then the parabola is shifted h units to the right.


If h < 0,


then the parabola is shifted h units to the left.


The constant k tells us how much the parabola is shifted up or down from the x-axis.


If k > 0,


then the parabola is shifted k units up.


If k < 0,


then the parabola is shifted k units down.


The constants h and k also tell us the coordinates of the vertex of the parabola.


The vertex is located at (h, k),


which means that we can easily find it by looking at


the equation in shifted form.


For example,


if we have y = 2(x - 3) + 5,


then we know that


the vertex is at (3, 5),


because h = 3 and k = 5.


How to Find


The Shifted Form of a Parabola


from


The Standard Form?




The standard form of a parabola


is another way of writing its equation.


It has


the following general form:


y = ax + bx + c


In this equation,


a, b, and c are constants that determine


the shape and position of


the parabola.


The constant a tells us how wide or narrow


the parabola is,


and whether it opens up or down,


just like in


the shifted form.


The constants b and c tell us how much


the parabola is shifted horizontally and vertically from its original position,


but in a less obvious way than in


the shifted form.


To find


the shifted form of a parabola from


the standard form,


we need to use


a process called completing


the square.


This process involves adding and subtracting


a certain term to


the standard form equation to make it look like


the shifted form equation.


Here are


the steps to follow:


Rewrite


How to Graph


The Shifted Form of


A Parabola Using Transformations?




To graph


The Shifted Form of


A Parabola using transformations,


we need to use


The Vertex Form


as our reference point.


The Vertex Form


is


The Shifted Form


when h = 0 and k = 0,


which means that


The Vertex


is at (0, 0) and


The Axis Of Symmetry


is


The Y-Axis.


The Vertex Form


has


The Equation:


y = ax


To graph


The Shifted Form,


we need to apply


two transformations:


A Horizontal Shift


and


A Vertical Shift.


A Horizontal Shift


moves


The Parabola


left or right by h units.


If h > 0,


then


The Parabola


moves h units to


The Right.


If h < 0,


then


The Parabola


moves h units to


The Left.


A Vertical Shift


moves


The Parabola


up or down by k units.


If k > 0,


then


The Parabola


moves k units up.


If k < 0,


then


The Parabola


moves k units down.


To graph


The Shifted Form,


we need to follow these steps:


Plot


The Vertex


at (h, k).


This is


The Highest Or Lowest Point


of


The Parabola,


depending on whether it opens up or down.





Draw


The Axis Of Symmetry


as


A Vertical Line


that passes through


The Vertex.


This line divides


The Parabola


into two equal parts.





Find


Two Other Points


on


One Side Of The Axis Of Symmetry


by plugging in values for x into


The Equation.


For example,


if we have y = 2(x - 3) + 5,


then we can plug in x = 4 and x = 5 to get y = 9 and y = 17,


respectively.


These points are (4, 9) and (5, 17).


Plot these points on


The Graph.





Use


Symmetry


to find


Two Other Points


on


The Other Side Of The Axis Of Symmetry.


For example,


if we have y = 2(x - 3) + 5,


then we can use x = 3 - (4 - 3) = 2 and x = 3 - (5 - 3) = 1 to get y = 9 and y = 17,


respectively.


These points are (2, 9) and (1, 17).


Plot these points on


The Graph.





Connect


The Points


with a smooth curve to form


The Parabola.


Make sure that the curve passes through


The Vertex


and is symmetrical about


The Axis Of Symmetry.





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