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**Learning Objectives**

In this section, you will:

- Recognize characteristics of parabolas.
- Understand how the graph of a parabola is related to itsquadratic function.
- Determine a quadratic function’s minimum or maximum value.
- Solve problems involving a quadratic function’s minimum ormaximum value.

**Figure****1**An array ofsatellite dishes. (credit: Matthew Colvin de Valle, Flickr)

Curved antennas, such as the ones shown in__Figure1__, are commonly used to focus microwaves and radio waves totransmit television and telephone signals, as well as satellite andspacecraft communication. The cross-section of the antenna is inthe shape of a parabola, which can be described by a quadraticfunction.

In this section, we will investigate quadratic functions, whichfrequently model problems involving area and projectile motion.Working with quadratic functions can be less complex than workingwith higher degree functions, so they provide a good opportunityfor a detailed study of function behavior.

**Recognizing Characteristics ofParabolas**

The graph of a quadratic function is a U-shaped curve calledaparabola. One important feature of the graph is that it hasan extreme point, called the**vertex**. If theparabola opens up, the vertex represents the lowest point on thegraph, or theminimum valueof the quadratic function. Ifthe parabola opens down, the vertex represents the highest point onthe graph, or themaximum value. In either case, the vertex isa turning point on the graph. The graph is also symmetric with avertical line drawn through the vertex, calledthe**axis of symmetry**. These features areillustrated in__Figure2__.

**Figure****2**

The*y*-intercept is the pointat which the parabola crosses the*y*-axis. The*x*-intercepts are the points at which the parabolacrosses the*x*-axis. If theyexist, the*x*-interceptsrepresentthe**zeros****,**or**roots**,of the quadratic function, the values ofxxatwhichy=0.y=0.

**EXAMPLE1**

**Identifying the Characteristics of aParabola**

Determine the vertex, axis of symmetry, zeros,andy-y-intercept of the parabola shownin__Figure3__.

**Figure****3**

**Answer**

**Understanding How the Graphs ofParabolas are Related to Their Quadratic Functions**

The**general form****of aquadratic function**presents the function in theform

f(x)=ax2+bx+cf(x)=ax2+bx+c

wherea,b,a,b,andccare real numbersanda≠0.a≠0.Ifa>0,a>0,the parabolaopens upward. Ifa<0,a<0,the parabola opensdownward. We can use the general form of a parabola to find theequation for the axis of symmetry.

The axis of symmetry is defined byx=−b2a.x=−b2a.Ifwe use the quadraticformula,x=−b±b2−4ac√2a,x=−b±b2−4ac2a,tosolveax2+bx+c=0ax2+bx+c=0forthex-x-intercepts, or zeros, we find the valueofxxhalfway between them isalwaysx=−b2a,x=−b2a,the equation for the axis ofsymmetry.

__Figure 4__represents the graph of the quadraticfunction written in general formasy=x2+4x+3.y=x2+4x+3.In thisform,a=1,b=4,a=1,b=4,andc=3.c=3.Becausea>0,a>0,theparabola opens upward. The axis of symmetryisx=−42(1)=−2.x=−42(1)=−2.This also makes sense becausewe can see from the graph that the verticallinex=−2x=−2divides the graph in half. The vertexalways occurs along the axis of symmetry. For a parabola that opensupward, the vertex occurs at the lowest point on the graph, in thisinstance,(−2,−1).(−2,−1).Thex-x-intercepts,those points where the parabola crosses thex-x-axis,occur at(−3,0)(−3,0)and(−1,0).(−1,0).

**Figure****4**

The**standard form of a quadraticfunction**presents the function in the form

f(x)=a(x−h)2+kf(x)=a(x−h)2+k

where(h,k)(h,k)is the vertex. Because the vertexappears in the standard form of the quadratic function, this formis also known as the**vertex form of a quadraticfunction**.

As with the general form, ifa>0,a>0,theparabola opens upward and the vertex is a minimum.Ifa<0,a<0,the parabola opens downward, and thevertex is a maximum.__Figure5__represents the graph of the quadratic functionwritten in standard formasy=−3(x+2)2+4.y=−3(x+2)2+4.Sincex–h=x+2x–h=x+2inthis example,h=–2.h=–2.In thisform,a=−3,h=−2,a=−3,h=−2,andk=4.k=4.Becausea<0,a<0,theparabola opens downward. The vertex is at(−2,4).(−2,4).

**Figure****5**

The standard form is useful for determining how the graph istransformed from the graph ofy=x2.y=x2.__Figure6__is the graph of this basic function.

**Figure****6**

Ifk>0,k>0,the graph shifts upward, whereasifk<0,k<0,the graph shifts downward.In__Figure5__,k>0,k>0,so the graph is shifted 4 unitsupward. Ifh>0,h>0,the graph shifts toward theright and ifh<0,h<0,the graph shifts to the left.In__Figure5__,h<0,h<0,so the graph is shifted 2 unitsto the left. The magnitude ofaaindicates the stretch ofthe graph. If|a|>1,|a|>1,the pointassociated with a particularx-x-value shifts fartherfrom the*x-*axis, so the graphappears to become narrower, and there is a vertical stretch. Butif|a|<1,|a|<1,the point associatedwith a particularx-x-value shifts closer tothe*x-*axis, so the graphappears to become wider, but in fact there is a verticalcompression. In__Figure5__,|a|>1,|a|>1,so the graphbecomes narrower.

The standard form and the general form are equivalent methods ofdescribing the same function. We can see this by expanding out thegeneral form and setting it equal to the standard form.

a(x−h)2+kax2−2ahx+(ah2+k)==ax2+bx+cax2+bx+ca(x−h)2+k=ax2+bx+cax2−2ahx+(ah2+k)=ax2+bx+c

For the linear terms to be equal, the coefficients must beequal.

–2ah=b,soh=−b2a–2ah=b,soh=−b2a

This is theaxis of symmetrywe defined earlier.Setting the constant terms equal:

ah2+kk====cc−ah2c−a−(b2a)2c−b24aah2+k=ck=c−ah2=c−a−(b2a)2=c−b24a

In practice, though, it is usually easier to rememberthat*k*is the output valueof the function when the inputish,h,sof(h)=k.f(h)=k.

**FORMSOF QUADRATIC FUNCTIONS**

A quadratic function is a polynomial function of degree two. Thegraph of aquadratic functionis a parabola.

The**general form of a quadraticfunction**isf(x)=ax2+bx+cf(x)=ax2+bx+cwherea,b,a,b,andccarereal numbers anda≠0.a≠0.

The**standard form of a quadraticfunction**isf(x)=a(x−h)2+kf(x)=a(x−h)2+kwherea≠0.a≠0.

The vertex(h,k)(h,k)is located at

h=–b2a,k=f(h)=f(−b2a)h=–b2a,k=f(h)=f(−b2a)

**HOWTO**

**Given a graph of a quadratic function, write theequation of the function in general form.**

- Identify the horizontal shift of the parabola; this valueish.h.Identify the vertical shift of the parabola; thisvalue isk.k.
- Substitute the values of the horizontal and vertical shiftforhhandk.k.in thefunctionf(x)=a(x–h)2+k.f(x)=a(x–h)2+k.
- Substitute the values of any point, other than the vertex, onthe graph of the parabola forxxandf(x).f(x).
- Solve for the stretch factor,|a|.|a|.
- Expand and simplify to write in general form.

**EXAMPLE2**

**Writing the Equation of a QuadraticFunction from the Graph**

Write an equation for the quadraticfunctionggin__Figure7__as a transformationoff(x)=x2,f(x)=x2,and then expand the formula, andsimplify terms to write the equation in general form.

**Figure****7**

**Answer**

**Analysis**

We can check our work using the table feature on a graphingutility. First enterY1=12(x+2)2−3.Y1=12(x+2)2−3.Next,selectTBLSET,TBLSET,thenuseTblStart=–6TblStart=–6andΔTbl=2,ΔTbl=2,andselectTABLE.TABLE.See__Table1__.

xx | –6 | –4 | –2 | 0 | 2 |

yy | 5 | –1 | –3 | –1 | 5 |

**Table****1**

The ordered pairs in the table correspond to points on thegraph.

**TRYIT#1**

A coordinate grid has been superimposed over the quadratic pathof a basketball in__Figure8__. Find an equation for the path of the ball. Does theshooter make the basket?

**Figure****8**(credit:modification of work by Dan Meyer)

**HOWTO**

**Given a quadratic function in general form, find thevertex of the parabola.**

- Identifya,b,andc.a,b,andc.
- Findh,h,the
*x*-coordinate of the vertex, bysubstitutingaaandbbintoh=–b2a.h=–b2a. - Findk,k,the
*y*-coordinate of the vertex, byevaluatingk=f(h)=f(−b2a).k=f(h)=f(−b2a).

**EXAMPLE3**

**Finding the Vertex of a QuadraticFunction**

Find the vertex of the quadraticfunctionf(x)=2x2–6x+7.f(x)=2x2–6x+7.Rewrite thequadratic in standard form (vertex form).

**Answer**

**Analysis**

One reason we may want to identify the vertex of the parabola isthat this point will inform us where the maximum or minimum valueof the output occurs,k,k,and where itoccurs,x.x.

**TRYIT#2**

Given the equationg(x)=13+x2−6x,g(x)=13+x2−6x,writethe equation in general form and then in standard form.

**Finding the Domain and Range of aQuadratic Function**

Any number can be the input value of a quadratic function.Therefore, the domain of any quadratic function is all realnumbers. Because parabolas have a maximum or a minimum point, therange is restricted. Since the vertex of a parabola will be eithera maximum or a minimum, the range will consist ofall*y*-values greater than orequal to the*y*-coordinate atthe turning point or less than or equal tothe*y*-coordinate at the turningpoint, depending on whether the parabola opens up or down.

**DOMAINAND RANGE OF A QUADRATIC FUNCTION**

The domain of anyquadratic functionis all realnumbers unless the context of the function presents somerestrictions.

The range of a quadratic function written in generalformf(x)=ax2+bx+cf(x)=ax2+bx+cwith apositiveaavalueisf(x)≥f(−b2a),f(x)≥f(−b2a),or[f(−b2a),∞);[f(−b2a),∞);therange of a quadratic function written in general form with anegativeaavalueisf(x)≤f(−b2a),f(x)≤f(−b2a),or(−∞,f(−b2a)].(−∞,f(−b2a)].

The range of a quadratic function written in standardformf(x)=a(x−h)2+kf(x)=a(x−h)2+kwith apositiveaavalue isf(x)≥k;f(x)≥k;the rangeof a quadratic function written in standard form with anegativeaavalue isf(x)≤k.f(x)≤k.

**HOWTO**

**Given a quadratic function, find the domain andrange.**

- Identify the domain of any quadratic function as all realnumbers.
- Determine whetheraais positive or negative.Ifaais positive, the parabola has a minimum.Ifaais negative, the parabola has a maximum.
- Determine the maximum or minimum value of theparabola,k.k.
- If the parabola has a minimum, the range is givenbyf(x)≥k,f(x)≥k,or[k,∞).[k,∞).Ifthe parabola has a maximum, the range is givenbyf(x)≤k,f(x)≤k,or(−∞,k].(−∞,k].

**EXAMPLE4**

**Finding the Domain and Range of aQuadratic Function**

Find the domain and rangeoff(x)=−5x2+9x−1.f(x)=−5x2+9x−1.

**Answer**

**TRYIT#3**

Find the domain and rangeoff(x)=2(x−47)2+811.f(x)=2(x−47)2+811.

**Determining the Maximum and MinimumValues of Quadratic Functions**

The output of the quadratic function at the vertex is themaximum or minimum value of the function, depending on theorientation of theparabola. We can see the maximum andminimum values in__Figure9__.

**Figure****9**

There are many real-world scenarios that involve finding themaximum or minimum value of a quadratic function, such asapplications involving area and revenue.

**EXAMPLE5**

**Finding the Maximum Value of aQuadratic Function**

A backyard farmer wants to enclose a rectangular space for a newgarden within her fenced backyard. She has purchased 80 feet ofwire fencing to enclose three sides, and she will use a section ofthe backyard fence as the fourth side.

- ⓐFind a formula for the area enclosed by the fence if the sidesof fencing perpendicular to the existing fence havelengthL.L.
- ⓑWhat dimensions should she make her garden to maximize theenclosed area?

**Answer**

**Analysis**

This problem also could be solved by graphing the quadraticfunction. We can see where the maximum area occurs on a graph ofthe quadratic function in__Figure11__.

**Figure****11**

**HOWTO**

**Given an application involving revenue, use a quadraticequation to find the maximum.**

- Write a quadratic equation for a revenue function.
- Find the vertex of the quadratic equation.
- Determine the
*y*-value ofthe vertex.

**EXAMPLE6**

**Finding Maximum Revenue**

The unit price of an item affects its supply and demand. Thatis, if the unit price goes up, the demand for the item will usuallydecrease. For example, a local newspaper currently has 84,000subscribers at a quarterly charge of $30. Market research hassuggested that if the owners raise the price to $32, they wouldlose 5,000 subscribers. Assuming that subscriptions are linearlyrelated to the price, what price should the newspaper charge for aquarterly subscription to maximize their revenue?

**Answer**

**Analysis**

This could also be solved by graphing the quadratic asin__Figure12__. We can see the maximum revenue on a graph of thequadratic function.

**Figure****12**

**Finding the***x*- and*y*-Intercepts of a Quadratic Function

*x*- and

*y*-Intercepts of a Quadratic Function

Much as we did in the application problems above, we also needto find intercepts of quadratic equations for graphing parabolas.Recall that we find they-y-intercept of a quadratic byevaluating the function at an input of zero, and we findthex-x-intercepts at locations where the output iszero. Notice in__Figure13__that the number ofx-x-intercepts canvary depending upon the location of the graph.

**Figure****13**Numberof*x*-intercepts of aparabola

**HOWTO**

**Given a quadratic functionf(x),f(x),findthey-y-and x-intercepts.**

- Evaluatef(0)f(0)to find the
*y*-intercept. - Solve the quadratic equationf(x)=0f(x)=0to findthe
*x*-intercepts.

**EXAMPLE7**

**Finding the***y*- and*x*-Intercepts of a Parabola

*y*- and

*x*-Intercepts of a Parabola

Find the*y*-and*x*-intercepts of thequadraticf(x)=3x2+5x−2.f(x)=3x2+5x−2.

**Answer**

**Analysis**

By graphing the function, we can confirm that the graph crossesthe*y*-axisat(0,−2).(0,−2).We can also confirm that the graphcrosses the*x*-axisat(13,0)(13,0)and(−2,0).(−2,0).See__Figure14__

**Figure****14**

**Rewriting Quadratics in StandardForm**

In__Example7__, the quadratic was easily solved by factoring. However,there are many quadratics that cannot be factored. We can solvethese quadratics by first rewriting them in standard form.

**HOWTO**

**Given a quadratic function, findthex-x-intercepts by rewriting in standardform**.

- Substituteaaandbbintoh=−b2a.h=−b2a.
- Substitutex=hx=hinto the general form of thequadratic function to findk.k.
- Rewrite the quadratic in standard formusinghhandk.k.
- Solve for when the output of the function will be zero to findthex-x-intercepts.

**EXAMPLE8**

**Finding the***x*-Intercepts of a Parabola

*x*-Intercepts of a Parabola

Find thex-x-intercepts of the quadraticfunctionf(x)=2x2+4x−4.f(x)=2x2+4x−4.

**Answer**

**Analysis**

We could have achieved the same results using the quadraticformula. Identifya=2,b=4a=2,b=4andc=−4.c=−4.

x=====−b±b2−4ac√2a−4±42−4(2)(−4)√2(2)−4±48√4−4±3(16)√4−1±3–√x=−b±b2−4ac2a=−4±42−4(2)(−4)2(2)=−4±484=−4±3(16)4=−1±3

So the*x*-intercepts occurat(−1−3–√,0)(−1−3,0)and(−1+3–√,0).(−1+3,0).

**TRYIT#4**

In a__TryIt__, we found the standard and general form for thefunctiong(x)=13+x2−6x.g(x)=13+x2−6x.Now findthe*y*-and*x*-intercepts (if any).

**EXAMPLE9**

**Applying the Vertexand***x*-Intercepts of aParabola

*x*-Intercepts of aParabola

A ball is thrown upward from the top of a 40 foot high buildingat a speed of 80 feet per second. The ball’s height above groundcan be modeled by theequationH(t)=−16t2+80t+40.H(t)=−16t2+80t+40.

- ⓐWhen does the ball reach the maximum height?
- ⓑWhat is the maximum height of the ball?
- ⓒWhen does the ball hit the ground?

**Answer**

**TRYIT#5**

A rock is thrown upward from the top of a 112-foot high cliffoverlooking the ocean at a speed of 96 feet per second. The rock’sheight above ocean can be modeled by theequationH(t)=−16t2+96t+112.H(t)=−16t2+96t+112.

- ⓐWhen does the rock reach the maximum height?
- ⓑWhat is the maximum height of the rock?
- ⓒWhen does the rock hit the ocean?

**MEDIA**

Access these online resources for additional instruction andpractice with quadratic equations.

**5.1 SectionExercises**

**Verbal**

__1__.

Explain the advantage of writing a quadratic function instandard form.

2.

How can the vertex of a parabola be used in solving real-worldproblems?

__3__.

Explain why the condition ofa≠0a≠0is imposed in thedefinition of the quadratic function.

4.

What is another name for the standard form of a quadraticfunction?

__5__.

What two algebraic methods can be used to find the horizontalintercepts of a quadratic function?

**Algebraic**

For the following exercises, rewrite the quadratic functions instandard form and give the vertex.

6.

f(x)=x2−12x+32f(x)=x2−12x+32

__7__.

g(x)=x2+2x−3g(x)=x2+2x−3

8.

f(x)=x2−xf(x)=x2−x

__9__.

f(x)=x2+5x−2f(x)=x2+5x−2

10.

h(x)=2x2+8x−10h(x)=2x2+8x−10

__11__.

k(x)=3x2−6x−9k(x)=3x2−6x−9

12.

f(x)=2x2−6xf(x)=2x2−6x

__13__.

f(x)=3x2−5x−1f(x)=3x2−5x−1

For the following exercises, determine whether there is aminimum or maximum value to each quadratic function. Find the valueand the axis of symmetry.

14.

y(x)=2x2+10x+12y(x)=2x2+10x+12

__15__.

f(x)=2x2−10x+4f(x)=2x2−10x+4

16.

f(x)=−x2+4x+3f(x)=−x2+4x+3

__17__.

f(x)=4x2+x−1f(x)=4x2+x−1

18.

h(t)=−4t2+6t−1h(t)=−4t2+6t−1

__19__.

f(x)=12x2+3x+1f(x)=12x2+3x+1

20.

f(x)=−13x2−2x+3f(x)=−13x2−2x+3

For the following exercises, determine the domain and range ofthe quadratic function.

__21__.

f(x)=(x−3)2+2f(x)=(x−3)2+2

22.

f(x)=−2(x+3)2−6f(x)=−2(x+3)2−6

__23__.

f(x)=x2+6x+4f(x)=x2+6x+4

24.

f(x)=2x2−4x+2f(x)=2x2−4x+2

__25__.

k(x)=3x2−6x−9k(x)=3x2−6x−9

For the following exercises, use thevertex(h,k)(h,k)and a point on thegraph(x,y)(x,y)to find the general form of the equationof the quadratic function.

26.

(h,k)=(2,0),(x,y)=(4,4)(h,k)=(2,0),(x,y)=(4,4)

__27__.

(h,k)=(−2,−1),(x,y)=(−4,3)(h,k)=(−2,−1),(x,y)=(−4,3)

28.

(h,k)=(0,1),(x,y)=(2,5)(h,k)=(0,1),(x,y)=(2,5)

__29__.

(h,k)=(2,3),(x,y)=(5,12)(h,k)=(2,3),(x,y)=(5,12)

30.

(h,k)=(−5,3),(x,y)=(2,9)(h,k)=(−5,3),(x,y)=(2,9)

__31__.

(h,k)=(3,2),(x,y)=(10,1)(h,k)=(3,2),(x,y)=(10,1)

32.

(h,k)=(0,1),(x,y)=(1,0)(h,k)=(0,1),(x,y)=(1,0)

__33__.

(h,k)=(1,0),(x,y)=(0,1)(h,k)=(1,0),(x,y)=(0,1)

**Graphical**

For the following exercises, sketch a graph of the quadraticfunction and give the vertex, axis of symmetry, and intercepts.

34.

f(x)=x2−2xf(x)=x2−2x

__35__.

f(x)=x2−6x−1f(x)=x2−6x−1

36.

f(x)=x2−5x−6f(x)=x2−5x−6

__37__.

f(x)=x2−7x+3f(x)=x2−7x+3

38.

f(x)=−2x2+5x−8f(x)=−2x2+5x−8

__39__.

f(x)=4x2−12x−3f(x)=4x2−12x−3

For the following exercises, write the equation for the graphedquadratic function.

40.

__41__.

42.

__43__.

44.

__45__.

**Numeric**

For the following exercises, use the table of values thatrepresent points on the graph of a quadratic function. Bydetermining the vertex and axis of symmetry, find the general formof the equation of the quadratic function.

46.

xx | –2 | –1 | 0 | 1 | 2 |

yy | 5 | 2 | 1 | 2 | 5 |

__47__.

xx | –2 | –1 | 0 | 1 | 2 |

yy | 1 | 0 | 1 | 4 | 9 |

48.

xx | –2 | –1 | 0 | 1 | 2 |

yy | –2 | 1 | 2 | 1 | –2 |

__49__.

xx | –2 | –1 | 0 | 1 | 2 |

yy | –8 | –3 | 0 | 1 | 0 |

__50__.

xx | –2 | –1 | 0 | 1 | 2 |

yy | 8 | 2 | 0 | 2 | 8 |

**Technology**

For the following exercises, use a calculator to find theanswer.

51.

Graph on the same set of axes thefunctionsf(x)=x2f(x)=x2,f(x)=2x2f(x)=2x2,andf(x)=13x2f(x)=13x2.

What appears to be the effect of changing the coefficient?

52.

Graph on the same set ofaxesf(x)=x2,f(x)=x2+2f(x)=x2,f(x)=x2+2andf(x)=x2,f(x)=x2+5f(x)=x2,f(x)=x2+5andf(x)=x2−3.f(x)=x2−3.Whatappears to be the effect of adding a constant?

__53__.

Graph on the same set ofaxesf(x)=x2,f(x)=(x−2)2,f(x−3)2f(x)=x2,f(x)=(x−2)2,f(x−3)2,andf(x)=(x+4)2.f(x)=(x+4)2.

What appears to be the effect of adding or subtracting thosenumbers?

54.

The path of an object projected at a 45 degree angle withinitial velocity of 80 feet per second is given by thefunctionh(x)=−32(80)2x2+xh(x)=−32(80)2x2+xwherexxisthe horizontal distance traveled andh(x)h(x)is theheight in feet. Use the TRACE feature of your calculator todetermine the height of the object when it has traveled 100 feetaway horizontally.

__55__.

A suspension bridge can be modeled by the quadraticfunctionh(x)=.0001x2h(x)=.0001x2with−2000≤x≤2000−2000≤x≤2000where|x||x|isthe number of feet from the center andh(x)h(x)is heightin feet. Use the TRACE feature of your calculator to estimate howfar from the center does the bridge have a height of 100 feet.

**Extensions**

For the following exercises, use the vertex of the graph of thequadratic function and the direction the graph opens to find thedomain and range of the function.

56.

Vertex(1,−2),(1,−2),opens up.

__57__.

Vertex(−1,2)(−1,2)opens down.

58.

Vertex(−5,11),(−5,11),opens down.

__59__.

Vertex(−100,100),(−100,100),opens up.

For the following exercises, write the equation of the quadraticfunction that contains the given point and has the same shape asthe given function.

60.

Contains(1,1)(1,1)and has shapeoff(x)=2x2.f(x)=2x2.Vertex is onthey-y-axis.

__61__.

Contains(−1,4)(−1,4)and has the shapeoff(x)=2x2.f(x)=2x2.Vertex is onthey-y-axis.

62.

Contains(2,3)(2,3)and has the shapeoff(x)=3x2.f(x)=3x2.Vertex is onthey-y-axis.

__63__.

Contains(1,−3)(1,−3)and has the shapeoff(x)=−x2.f(x)=−x2.Vertex is onthey-y-axis.

64.

Contains(4,3)(4,3)and has the shapeoff(x)=5x2.f(x)=5x2.Vertex is onthey-y-axis.

__65__.

Contains(1,−6)(1,−6)has the shapeoff(x)=3x2.f(x)=3x2.Vertex has x-coordinateof−1.−1.

**Real-World Applications**

66.

Find the dimensions of the rectangular dog park producing thegreatest enclosed area given 200 feet of fencing.

__67__.

Find the dimensions of the rectangular dog park split into 2pens of the same size producing the greatest possible enclosed areagiven 300 feet of fencing.

68.

Find the dimensions of the rectangular dog park producing thegreatest enclosed area split into 3 sections of the same size given500 feet of fencing.

__69__.

Among all of the pairs of numbers whose sum is 6, find the pairwith the largest product. What is the product?

70.

Among all of the pairs of numbers whose difference is 12, findthe pair with the smallest product. What is the product?

__71__.

Suppose that the price per unit in dollars of a cell phoneproduction is modeledbyp=$45−0.0125x,p=$45−0.0125x,wherexxis inthousands of phones produced, and the revenue represented bythousands of dollars isR=x⋅p.R=x⋅p.Find the productionlevel that will maximize revenue.

72.

A rocket is launched in the air. Its height, in meters above sealevel, as a function of time, in seconds, is givenbyh(t)=−4.9t2+229t+234.h(t)=−4.9t2+229t+234.Find themaximum height the rocket attains.

__73__.

A ball is thrown in the air from the top of a building. Itsheight, in meters above ground, as a function of time, in seconds,is given byh(t)=−4.9t2+24t+8.h(t)=−4.9t2+24t+8.How longdoes it take to reach maximum height?

74.

A soccer stadium holds 62,000 spectators. With a ticket price of$11, the average attendance has been 26,000. When the price droppedto $9, the average attendance rose to 31,000. Assuming thatattendance is linearly related to ticket price, what ticket pricewould maximize revenue?

__75__.

A farmer finds that if she plants 75 trees per acre, each treewill yield 20 bushels of fruit. She estimates that for eachadditional tree planted per acre, the yield of each tree willdecrease by 3 bushels. How many trees should she plant per acre tomaximize her harvest?