# 12 - graphing quadratic eqations

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• 8/14/2019 12 - Graphing Quadratic Eqations

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Chapter 12

W

hile the straight lines

of linear equations

are easy to graph,they are not the most exciting.

however, opens the door to the

wonderful world of parabolas.

Chapter 12 introduces these

U-shaped curves and givesyou the tools to graph them.

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In this Chapter...Introduction to Parabolas

Write a Quadratic Equation in Vertex Form

Graph a Parabola

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Introduction to Parabolas

230

Steepness or Flatness of a ParabolaParabola Basics

When you graph a

quadratic equation,you will end up witha U-shaped curveknown as a parabola.

Note: A quadraticequation is an equationwhose highest exponentin the equation is two,such as x 2.

The standard way to write

a quadratic equation youwant to graph is in the formy = a(x h ) 2 + k , where a , hand k are numbers. Writinga quadratic equation in thisform provides information tohelp you graph the parabola.

Note: To write a quadraticequation in the formy = a (x h) 2 + k , see page 232.

The number represented

by a indicates thesteepness or flatness ofa parabola. The largerthe number, whetherthe number is positiveor negative, the steeperthe parabola.

The number represented

by a also indicates if aparabola opens upwardor downward. When a isa positive number, theparabola opens upward.When a is a negativenumber, the parabolaopens downward.

Remember being a kid and throwing that baseball

as high as you could? The ball never landed inthe same spot from which you tossed it because,

whether you noticed it or not, the ball always

followed a curve up and then fell back down on a

similar curve. Those curves formed a symmetrical,

U-shaped curve called a parabola.

When you graph a quadratic equationan

equation in which the highest exponent is

two, such as x2

the graph of the equation isa parabola. Parabolas open either downward

or upward.

The lowest point in an upward-opening parabola

or the highest point in a downward-openingparabola is called the vertex.

While quadratic equations can be expressed

in the form y= ax2 + bx+ c, for the purposes ofgraphing, parabolas are generally rewritten as

y= a(x h)2 + k. This is called the vertex form

of an equation. In this form, the variable a

determines how steep a parabola will be and

whether it will open upward or downward.The variables h and k represent the position

of the vertex (h,k) in the coordinate plane.

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2

1

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5

-1-2-3-4-5 2 3 4 5

y= 2(x 1)2 1

1

-1

-2

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-4

-5

2

1

3

4

5

-1-2-3-4-5 2 3 4 5

y= (x 1)2 11

2

y= (x 1)2 11

2

y = 2(x 1)2 1

y= 2(x 1)2

1

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231

Vertex of a Parabola

PracticeTip

1

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2

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3

4

5

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Determine the vertex of eachof the following parabolas andindicate if each parabola opens

1) y = (x 1) 2

2) y = (x )2 5

3) y = 3(x + 1) 2 + 2

4) y = 4x 2 +

5) y = x 2 + 2

6) y = (x 2)2 3

Can the graph of a quadraticequation with just one termbe a parabola?

Yes. The graphs of eventhe simplest quadraticequations, which bydefinition must contain thex2 variable, form a parabola.For example, a graph ofthe equation y= x2 formsa parabola and can bewritten as y= 1(x 0)2 + 0in the vertex form. In thisequation, you will noticethat a = 1, so the parabolaopens upward, and becauseh and k both equal 0, thevertex is at the origin (0,0)in the coordinate plane.

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2

1

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y= 2(x + 3)2 2

y= 2(x 1)2 1

y= 2(x 3)2+ 2

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2

1

3

4

5

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y= 2(x + 3)2 1 y= 2(x 2)

2 1

y= 2(x + 3)2+ 2 y= 2(x 2)

2+ 2

x-axis

y-axis

The numbers

represented by hand k , when writtenas (h,k ) , indicatethe location of thevertex of a parabola,which is the lowestor highest point of aparabola.

The number represented by h

indicates how far left or rightalong the x-axis the vertex ofa parabola is located from theorigin. The origin is the locationwhere the x-axis and y-axisintersect.

Note: When a minus sign () appearsafter x in the equation, the parabolamoves to the right. When a plus sign (+)appears after x in the equation, the

parabola moves to the left.

The number

represented by kindicates how far upor down along they-axis the vertex ofa parabola is locatedfrom the origin.

Note: Positive knumbers move theparabola up. Negativek numbers move the

parabola down.

Note: If a minus sign () appearsafter

xin the equation,

hin

(h,k) is a positive number. Ifa plus sign (+) appears after xin the equation, h in (h,k ) is anegative number.

For example, in theequation y = 2(x + 3) 2 2 ,the vertex of the parabola islocated at (3 , 2) .

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Write a QuadraticEquation in Vertex Form

232

Standard Form Example

y = ax 2 + bx + c y = 2x 2 4x + 3

Vertex Form Example

y = a (x h )2 + k y = 2(x 1) 2 + 1

are often writtenin the standard formy = ax 2 + bx + c ,where a , b and c arenumbers and one sideof the equation is setto equal y .

Note: A quadratic equationis an equation whosehighest exponent in the

equation is two, such as x2

.

When you want to graph

a quadratic equation, youwill want to write theequation in the vertexform y = a (x h) 2 + k ,where a , h and k arenumbers. Writing aquadratic equation in thisform provides informationto help you graph theequation.

equation from the standardform to the vertex form, youfirst need to identify thenumbers a , b and c in thestandard form of the equation.

Note: If the x variable or cnumber does not appear inthe equation, assume the numberis 0 . For example, the equation

y = 2x 2 4x is the same as

y = 2x2

4x + 0 .

To determine the

value of a in thevertex form of theequation, use thevalue of a from thestandard form ofthe equation, sincethe numbers will bethe same.

In this example, aequals 2 .

1 2

Graphing a parabola is simple if you first change

the quadratic equation from standard form(y= ax2 + bx+ c) to vertex form (y= a(x - h)2 + k).

The vertex form of a quadratic equation gives you

the direction that the parabola opens, as well as

the coordinates of the vertexthe highest point

of a parabola that opens downward or lowest

point of a parabola that opens upward.To convert a quadratic equation to vertex form,

you must determine the values of h and kthe

value of a is the same in both equations. To

determine the value of h, you use a simpleformula (h = ) that uses the standard form

values of a and b. To determine the value of k,

go back to the standard form of the equation.

Replace the variable ywith the variable k and

replace the xvariables with the value of h. You

can then solve for the variable k in the equation.When you have all three valuesa, h and kyou can write the equation out in vertex form.

b2a

Standard Form Example

y = ax 2 + bx + c y = 2x 2 4x + 3

a = 2, b = 4, c = 3

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233

PracticeTip

1) y = x 2 2x + 3

2) y = x 2 + 4x 1

3) y = 2x 2 4x 3

4) y = x 2 6x + 7

5) y = x 2 + 3

6) y = x 2 + 10x + 15

How do I change an equation from vertex formto the standard form of a quadratic equation?

If you multiply and then simplify the terms of

an equation in vertex form, you will arrive atthe standard form of the quadratic equation.This is a great way to check your answer afterconverting an equation from standard form tovertex form. For information on multiplyingpolynomials, see page 154.

3 4 5To determine thevalue of

hin the

vertex form of theequation, use thenumbers a and b fromthe standard form ofthe equation in theformula h = .

In this example, hequals 1 .

In the vertex formof the equation,replace the numbersyou determined fora , h and k .

You have finishedchanging the quadraticequation from thestandard form to thevertex form.

To determine the valueof

k, use the standard

form of the equation andreplace the variable y withthe variable k and replacethe x variables with thevalue of h you determinedin step 3. Then solve for kin the equation.

In this example, k equals 1 .

b2a

h =

= = = 14

4(4)

2 x 2

b2a

k = 2h 2 4h + 3

= 2(1)2 4(1) + 3

= 2 4 + 3

= 1

y = a (x h )2 + k

y = 2(x 1)2 + 1

y = 2(x 1) 2 + 1

y = 2(x 1)(x 1) + 1

y = 2(x2

2x + 1) + 1y = 2x 2 4x + 2 + 1

y = 2x 2 4x + 3

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Graph the equationy = 2(x 1)2 3.

y = a (x h )2 + k

y = 2(x 1)2 3

Location of parabolas vertex:(1,3)

To determine the location ofthe vertex of the parabola,which is the lowest or highestpoint of the parabola, look atthe h and k numbers in theequation.

Write the h and k numbers asan ordered pair in the form(h ,k ) . An ordered pair is twonumbers, written as (x ,y) ,which gives the location of a

point in the coordinate plane.

Note: If a minus sign ( )appears after x in theequation, the value of hin (h ,k ) is a positivenumber. If a plus sign (+)appears after x in theequation, the value of hin (h ,k ) is a negativenumber.

Plot the point forthe parabolasvertex in thecoordinate plane.

Note: To plot pointsin a coordinateplane, see page 82.

To determine if the parabola willopen upward or downward, lookat the number represented by a .When the value of a is a positivenumber, the parabola will openupward. When the value of a isa negative number, the parabolawill open downward.

In the equationy = 2(x 1) 2 3 ,the parabola will open upwardsince the value of a , or 2 , is a

positive number.

3 41

2

When you graph a quadratic equation in the form

ofy= ax2 + bx+ c in the coordinate plane, you willalways end up with a U-shaped curve known as a

parabola. For information on the coordinate

plane, see page 80.

When you want to graph a quadratic equation, it

is useful to have the equation written in vertex

form (y= a(x h)2 + k). The vertex form tells you

whether the parabola opens upward or downwardand provides you with the coordinates of the

parabola's vertexthe highest or lowest point of

the parabola. In an equation in vertex form, if

the value of a is positive, the parabola opensupward, while if the value of a is negative, the

parabola opens downward. The values h and k

represent the coordinates (h,k) of the parabolas

vertex.

After you plot the vertex, plotting one point oneither side of the vertex is sufficient to draw the

rest of the curve. If you want to make sure youdid not make a mistake, you can plot more

points.

Graph a Parabola

234

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x-axis

y-axis

(1,3)

y= 2(x 1)2 3

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PracticeTip

Graph the following parabolas. You

can check your graphs on page 265.1) y = (x 1) 2 + 3

2) y = (x + 2) 2 4

3) y = x 2 1

4) y = x 2 + 3

5) y = (x 2) 2 1

6) y = 2(x + 3) 2

Can graphing a parabola help me solve

a quadratic equation?Yes. Look at where a parabola crossesthe x-axis. The values forxat thosepoints will give you the solutions tothe quadratic equation. If the parabolacrosses the x-axis at two points, theequation has two solutions. If theparabola touches the x-axis at just onepoint, the equation has one solution.If the parabola does not cross the x-axis,the equation has no real solutions.

Choose a randomnumber for the xvariable. For example,let x equal 0 .

Place the number youselected into theequation to determinethe value of the yvariable. Then solvefory in the equation.

Plot the two points ina coordinate plane.

Connect the points todraw a smooth curve.Draw an arrow at eachend of the curve toshow that the parabolaextends forever.

The parabola shows allthe possible solutionsto the equation.

Write the x andy valuetogether as an orderedpair in the form (x ,y ) .

Repeat steps 5 to 7to determine anotherordered pair. The orderedpair should be locatedon the other side of theparabolas vertex so youhave one point on eitherside of the parabolas

vertex.

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108

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x-axis

y-axis

(0,1) (2,1)

y= 2(x 1)2 3

Letx = 0

y = 2(x 1) 2 3

y = 2(0 1)2

3y = 2 3

y = 1

Ordered pair(x,y) = (0,1)

Letx = 2

y = 2(x 1) 2 3

y = 2(2 1)2 3

y = 2 3

y = 1

Ordered pair(x,y) = (2,1)

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236

Determine the vertex of the following parabolasand whether the parabolas open upward or

downward.

Question 1.

a) y = (x 1)2

+ 2

b) y = (x + 6)2

3

c) y = 4(x + 4)2

+

d) y = (x )

2

e) y = 200(x 50)2

75

Write the following quadratic equations in vertexform. Determine the vertex of each parabola andwhether the parabola opens upward or downward.

Question 2.

a) y = x2

2x + 1

b) y = x2

+ 6x + 5

c) y = x2

2x + 2

d) y = 4x2

+ 4x + 3

e) y = 25x2

+ 20x + 16

12

12

13

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237

Graph the following parabolas.Question 3.

a) y = (x + 1)

2

2

b) y = (x 3)2

+ 1

c) y = 2x2

8x + 5

d) y = x2

+ 4x

e) y = 2x2

12x 14