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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