# 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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