CHAPTER 2 The Metric System ConversionsMeasurement Significant Digits Graphing.

CHAPTER 2CHAPTER 2

The Metric SystemThe Metric System

ConversionsConversions

MeasurementMeasurement

Significant DigitsSignificant Digits

GraphingGraphing

Measure The Room LabMeasure The Room Lab

Measure the length and width of the room Measure the length and width of the room in “shoe” units. in “shoe” units. Discuss differences.Discuss differences.Why doe we use standards?Why doe we use standards?Name US units volume, distance, …etcName US units volume, distance, …etcName Metric “Base” units.Name Metric “Base” units.Name Metric prefixes Name Metric prefixes Place prefixes in order smallest to largest.Place prefixes in order smallest to largest.

2.1 The Metric System and SI2.1 The Metric System and SI

Why use the Metric SystemWhy use the Metric System

Based on powers of 10, convenient to useBased on powers of 10, convenient to use

The Syste’me Interationale d’Unite’s =The Syste’me Interationale d’Unite’s =

SI = The Metric SystemSI = The Metric System

This is the standard system used This is the standard system used throughout the world by scientists, throughout the world by scientists, engineers, and everyone else everywhere engineers, and everyone else everywhere except US. except US.

Why don’t we use the Metric Why don’t we use the Metric System in the United States?System in the United States?

Good question.Good question.

Base UnitsBase Units

Length = meter (m)Length = meter (m)

Mass = gram (g) Mass = gram (g) → “standard” unit = kg→ “standard” unit = kg

Volume = liter (l)Volume = liter (l)

Time = second (s)Time = second (s)

Temperature = Kelvin (K)Temperature = Kelvin (K)

Amount of a substance = mole (mol)Amount of a substance = mole (mol)

……etcetc

Definition of a kilogramDefinition of a kilogram

The mass of a small The mass of a small platinum-iridium metal platinum-iridium metal cylinder kept at a very cylinder kept at a very controlled temperature controlled temperature and humidity.and humidity.

Definition of a meterDefinition of a meter

The distance traveled by light in a vacuum The distance traveled by light in a vacuum during a time interval of 1/299 792 458 during a time interval of 1/299 792 458 seconds.seconds.

Definition of a secondDefinition of a second

The fraquency of one type of radiation The fraquency of one type of radiation emitted by a cesium-133 atom.emitted by a cesium-133 atom.

SI PrefixesSI Prefixes

Giga – GGiga – G 101099 1 000 000 0001 000 000 000Mega –MMega –M 101066 1 000 0001 000 000kilo – kkilo – k 101033 10001000hecto – hhecto – h 101022 100100deka – dadeka – da 1010 1010BASE UNITBASE UNIT meters/liters/grams/…etcmeters/liters/grams/…etcdeci – ddeci – d 1010-1-1 0.10.1centi – ccenti – c 1010-2-2 0.010.01milli – mmilli – m 1010-3-3 0.0010.001micro – micro – μμ 1010-6-6 0.000 0010.000 001nano – nnano – n 1010-9-9 0.000 000 0010.000 000 001pico – ppico – p 1010-12-12 0.000 000 000 0010.000 000 000 001

Introduction to meter stickIntroduction to meter stick

Meter (m) – dm – cm - mmMeter (m) – dm – cm - mm

METRIC “STEP” SYSTEM & METRIC “STEP” SYSTEM & CONVERSIONSCONVERSIONS

Convert 102m Convert 102m → mm→ mm

102m = 102 000mm102m = 102 000mm

Convert 427 693Convert 427 693μμm → mm → m

427 693427 693μμm = 0.427 693mm = 0.427 693m

Metric Conversion Hand Out #1 Metric Conversion Hand Out #1

METRIC “Step METRIC “Step System”System”

G

M

k

dah

BASE

d

mc

n

p

G

M

k

dah

BASE

d

mc

n

p

G

M

k

dah

BASE

d

mc

n

p

For each step you go up, move the decimal point one place to the left.

For each step you go down, move the decimal point one place to the right.

μ

Conversions Cont.Conversions Cont.

LengthLength 1 inch = 2.54 cm (exactly)1 inch = 2.54 cm (exactly)

VolumeVolume 1 liter = 1.0576 qt1 liter = 1.0576 qt

MassMass 1 kg = 2.21 lbs1 kg = 2.21 lbs

WeightWeight 1 lb = 4.45N (Newtons)1 lb = 4.45N (Newtons)

More useful conversions on back cover of More useful conversions on back cover of text booktext book

Dimensional Analysis MethodDimensional Analysis Method= Factor Label Method= Factor Label Method

Example #1Example #1

4km 4km → in→ in

4km4km x x 1000m1000m x x 100cm100cm x x 1in 1in = =

1 1km 1m 2.54cm1 1km 1m 2.54cm

157 480in157 480in

Example #2Example #2

26dam 26dam → yds→ yds

26dam26dam x x 1000cm1000cm x x 1in1in x x 1ft1ft x x 1yd1yd = =

1 1dam 2.54cm 12in 3ft1 1dam 2.54cm 12in 3ft

284.3yds284.3yds

More Example ProblemsMore Example Problems

3) 37hl 3) 37hl → gal→ gal

37hl = 978.28gal37hl = 978.28gal

4) 439 672 107mg → tons4) 439 672 107mg → tons

439 672 107mg = 0.4858tons439 672 107mg = 0.4858tons

5) 467 223 921 732 oz 5) 467 223 921 732 oz → Gg→ Gg

467 223 921 732 oz = 13,269.16Gg467 223 921 732 oz = 13,269.16Gg

6) 937 456 737mg 6) 937 456 737mg → tons→ tons

937 456 737mg = 1.03tons937 456 737mg = 1.03tons

Hint: Hint:

Insert the units first to ensure that the units Insert the units first to ensure that the units will cancel out leaving only the unit that will cancel out leaving only the unit that you want to end up with.you want to end up with.

Handouts #3 & #4Handouts #3 & #4

When using the dimensional analysis When using the dimensional analysis method it is very helpful to insert units first, method it is very helpful to insert units first, then the proper numbers.then the proper numbers.

SCIENTIFIC NOTATIONSCIENTIFIC NOTATION(and Calculators)(and Calculators)

Convert 276Gl Convert 276Gl → pl→ pl

276Gl = 276000000000000000000000pl276Gl = 276000000000000000000000pl

Convert 146ng → MgConvert 146ng → Mg

146ng = 0.000000000000146Mg146ng = 0.000000000000146Mg

Q: Is it convenient to use these Q: Is it convenient to use these types of numbers?types of numbers?

A: NO!!!!!!A: NO!!!!!!

Scientific Notation is used to Scientific Notation is used to represent these very large/small represent these very large/small numbers.numbers.

Rules for Scientific NotationRules for Scientific Notation

The numerical part of the quantity is The numerical part of the quantity is written as a number between 1 and 10 written as a number between 1 and 10 multiplied by a whole-number power of 10. multiplied by a whole-number power of 10.

M = 10M = 10nn where: 1 where: 1 ≤ M < 10 ≤ M < 10

n is an integern is an integer

If the decimal point must be moved to the If the decimal point must be moved to the right to achieve 1 right to achieve 1 ≤ M < 10, then n is ≤ M < 10, then n is negative (-).negative (-).

If the decimal point must be moved to the If the decimal point must be moved to the left to achieve 1 left to achieve 1 ≤ M < 10, then n is ≤ M < 10, then n is positive (+).positive (+).

101000 = 1 = 1

Therefore written in proper Therefore written in proper scientific notation:scientific notation:

276000000000000000000000 pl 276000000000000000000000 pl = 2.76 x 10 = 2.76 x 102323plpl

0.000000000000146ng = 1.46 x 100.000000000000146ng = 1.46 x 10-13-13MgMg

Calculator ButtonsCalculator Buttons

In class examples of E, EE, and In class examples of E, EE, and positive/negative exponents.positive/negative exponents.

Addition & SubtractionAddition & Subtraction

If the numbers have the same exponent, If the numbers have the same exponent, n, add or subtract the values of M and n, add or subtract the values of M and keep the same n.keep the same n.

3.7 x 103.7 x 1044 + 6.2 x 10 + 6.2 x 1044

= (3.7 + 6.2) x 10= (3.7 + 6.2) x 1044

= 9.9 x 10= 9.9 x 1044

Example-2Example-2

9.3 x 109.3 x 1077 - 4.1 x 10 - 4.1 x 1077

= (9.3 – 4.1) x 10= (9.3 – 4.1) x 1077

= 5.2 x 10= 5.2 x 1077

If the exponents are not the same, move If the exponents are not the same, move the decimal point to the left or right until the decimal point to the left or right until the exponents are the same. Then add or the exponents are the same. Then add or subtract M.subtract M.

Example-1Example-1

2.1 x 102.1 x 108 8 + 7.9 x 10 + 7.9 x 1055

= 2.1 x 10= 2.1 x 108 8 + 0.0079 x 10 + 0.0079 x 1088

= (2.1 + 0.0079) x 10= (2.1 + 0.0079) x 1088

= 2.1079 x 10= 2.1079 x 1088

oror

Example – 2Example – 2

2.1 x 102.1 x 1088 + 7.9 x 10 + 7.9 x 1055

= 2100 x 10= 2100 x 1055 + 7.9 x 10 + 7.9 x 1055

= (2100 + 7.9) x 10= (2100 + 7.9) x 1055

= 2107.9 x 10= 2107.9 x 1055

= 2.1079 x 10= 2.1079 x 1088

Exactly the same as previous exampleExactly the same as previous example

If the magnitude of one number is very If the magnitude of one number is very small compared to the other number, its small compared to the other number, its effect on the larger number is insignificant. effect on the larger number is insignificant. The smaller number can be treated as The smaller number can be treated as zero. (9.99 x 10zero. (9.99 x 1033 = 9999) = 9999)

7.98 x 107.98 x 1012 12 - 9.99 x 10 - 9.99 x 1033

= 7980000000 x 10= 7980000000 x 1033 - 9.99 x 10 - 9.99 x 1033

= (7980000000 - 9.99) x 10= (7980000000 - 9.99) x 1033

= 7979999990.01 x 10= 7979999990.01 x 1033

= 7.98 x 10= 7.98 x 101212

MultiplicationMultiplication

Multiply the values of M and add the Multiply the values of M and add the exponents, n. Multiply the units.exponents, n. Multiply the units.

4.37 x 104.37 x 1077m x 6.17 x 10m x 6.17 x 101313ss

= (4.37 x 6.17) x 10 = (4.37 x 6.17) x 10 (7 + 13)(7 + 13) (m x s) (m x s)

= 26.9629 x 10= 26.9629 x 102020msms

= 2.69629 x 10= 2.69629 x 102121msms

DivisionDivision

Divide the values of M and subtract the Divide the values of M and subtract the exponents of the divisor from the exponent of exponents of the divisor from the exponent of the dividend. Divide the units.the dividend. Divide the units.7.9 x 107.9 x 1099 m m44

3.1 x 103.1 x 1066 m m33

7.97.9= 3.1 x 10 = 3.1 x 10 (9 -6)(9 -6) m m (4-3)(4-3)

= 2.548 x 10= 2.548 x 1033mm

Challenging AdditionChallenging Addition

8.9 x 108.9 x 1055m + 7.6 10m + 7.6 1033kmkm

= 8.9 x 10= 8.9 x 1055m + 7600 x 10m + 7600 x 1033mm

= 8.9 x 10= 8.9 x 1055m + 76 x 10m + 76 x 1055mm

= (8.9 + 76) x 10= (8.9 + 76) x 1055mm

= 84.9 x 10= 84.9 x 1055mm

= 8.49 x 10= 8.49 x 1066mm

oror

Challenging Addition Cont.Challenging Addition Cont.

8.9 x 108.9 x 1055m + 7.6 x 10m + 7.6 x 1033kmkm

= 0.0089 x 10= 0.0089 x 1055km + 7.6 x 10km + 7.6 x 1033kmkm

= 0.89 x 10= 0.89 x 1033km + 7.6 x 10km + 7.6 x 1033kmkm

= 8.49 x 10= 8.49 x 1033kmkm

8.49 x 108.49 x 1033km = 8.49 x 10km = 8.49 x 1066mm

Challenging MultiplicationChallenging Multiplication

2.7 x 102.7 x 101010μμl X 4.3 x 10l X 4.3 x 10-4-4clcl

= 0.00027 x 10= 0.00027 x 101010cl X 4.3 x 10cl X 4.3 x 10-4-4clcl

= (0.00027 x 4.3) x 10 = (0.00027 x 4.3) x 10 (10-4)(10-4) (cl x cl) (cl x cl)

= 0.001161 x 10= 0.001161 x 1066clcl22

= 1.161 x 10= 1.161 x 1033clcl22

Challenging DivisionChallenging Division6.2 x 106.2 x 1088kgkg

4.2 x 104.2 x 10-5-5MgMg

6.2 x 106.2 x 1088kgkg

= 4200 x 10= 4200 x 10-5-5kgkg

6.2 6.2

= 4200 x 10 = 4200 x 10 (8- -5)(8- -5)

= 0.00147 x 10= 0.00147 x 101313

= 1.47 x 10= 1.47 x 101010

SECTION 2.2SECTION 2.2

Measurement UncertaintiesMeasurement Uncertainties

Comparing ResultsComparing Results

Three students measure the width of a sheet of paper Three students measure the width of a sheet of paper multiple times.multiple times.

#1 18.5cm#1 18.5cm→19.1cm, avg=18.8cm →19.1cm, avg=18.8cm ∴(18.8 ± 0.3)cm∴(18.8 ± 0.3)cm#2 18.8cm→19.2cm, avg=19.0cm ∴(19.0 ± 0.2)cm#2 18.8cm→19.2cm, avg=19.0cm ∴(19.0 ± 0.2)cm#3 18.2cm→18.4cm, avg=18.3cm ∴(18.3 ± 0.1)cm#3 18.2cm→18.4cm, avg=18.3cm ∴(18.3 ± 0.1)cmQ: Are the three measurements in agreement?Q: Are the three measurements in agreement?A: Students #1 & #2 have measurements that overlap, A: Students #1 & #2 have measurements that overlap,

both have measurements between 18.8cm→19.1cmboth have measurements between 18.8cm→19.1cm∴ ∴ #1 and #2 are in agreement.#1 and #2 are in agreement.

However, student #3 does not have any overlap with However, student #3 does not have any overlap with #1 or #2, ∴ there is no agreement between student #1 or #2, ∴ there is no agreement between student #3 and/or #1 & #2.#3 and/or #1 & #2.

Accuracy and PrecissionAccuracy and Precission

Precision =Precision =

The degree of exactness with which a The degree of exactness with which a quantity is measured using a given quantity is measured using a given instrument.instrument.

Q: Which student had the most precise Q: Which student had the most precise measurement?measurement?

A: #3 18.2cm–18.4cm, all measurements A: #3 18.2cm–18.4cm, all measurements are within are within ± 0.1cm.± 0.1cm.

Generally when measuring quantities, the Generally when measuring quantities, the device that has the finest divisions on its device that has the finest divisions on its scale yields the most precise scale yields the most precise measurement.measurement.

The precision of a measurement is ½ the The precision of a measurement is ½ the smallest division of the instrument.smallest division of the instrument.

Q: How precise is a meter stick?Q: How precise is a meter stick?

A: The smallest division on a meter stick is A: The smallest division on a meter stick is a millimeter(mm) a millimeter(mm) you can measure an you can measure an object to within 0.5 mm.object to within 0.5 mm.

Accuracy =Accuracy =

How well the results of an experiment or How well the results of an experiment or measurement agree with an accepted measurement agree with an accepted standard value.standard value.

If the accepted/standard value of the sheet If the accepted/standard value of the sheet of paper was 19.0cm wide, which student of paper was 19.0cm wide, which student was the most accurate, least accurate?was the most accurate, least accurate?

Most accurate = #2.Most accurate = #2.

Least accurate = #3.Least accurate = #3.

When checking the accuracy of a When checking the accuracy of a measuring device use the Two-Point measuring device use the Two-Point calibration method.calibration method.

#1 Make sure the instrument reads 0 when #1 Make sure the instrument reads 0 when it should.it should.

#2 Make sure the instrument yields the #2 Make sure the instrument yields the correct measurement on some accepted correct measurement on some accepted standard.standard.

Techniques of Good Techniques of Good MeasurementsMeasurements

Measurements must be made carefully.Measurements must be made carefully.

Common source of error = reading an Common source of error = reading an instrument when looking at it from an instrument when looking at it from an angle angle read the instrument from directly read the instrument from directly above.above.

Parallax = the apparent shift in position of Parallax = the apparent shift in position of an object when viewed from different an object when viewed from different angles.angles.

Significant digitsSignificant digits

Significant Digits = the valid digits in a Significant Digits = the valid digits in a measurement.measurement.

The last (estimated) digit is called the The last (estimated) digit is called the uncertain digit.uncertain digit.

All non zero digits in a measurement are All non zero digits in a measurement are significant.significant.

A = 1.24m B = 0.23cmA = 1.24m B = 0.23cm

How many significant digits for A & B?How many significant digits for A & B?

A = 3 B = 2A = 3 B = 2

Which is a more precise measurement?Which is a more precise measurement?

A is to the nearest cmA is to the nearest cm

B is to the nearest 1/100cmB is to the nearest 1/100cm

B is the more precise measurementB is the more precise measurement

ZEROSZEROS

Q: Are all zeros significant?Q: Are all zeros significant?

A: NoA: No

Q: Which zeros are significant?Q: Which zeros are significant?

0.0860m # of significant digits =?0.0860m # of significant digits =?

A: 3 significant digits, first 2 zeros only A: 3 significant digits, first 2 zeros only show the decimal place, the last one is show the decimal place, the last one is significant, it indicates the degree of significant, it indicates the degree of precision of the measuring device.precision of the measuring device.

186 000 m186 000 m

Q: How many significant digits?Q: How many significant digits?

A: ???????????? Cannot tell, it is A: ???????????? Cannot tell, it is ambiguous, you do not know what ambiguous, you do not know what instrument was used to achieve this instrument was used to achieve this measurement, possibly 3, 4, 5 or 6 measurement, possibly 3, 4, 5 or 6 significant digits.significant digits.

To avoid confusion rewrite #To avoid confusion rewrite #

186 km 186 km = 3 sig dig= 3 sig dig

186.000 km186.000 km = 6 sig dig= 6 sig dig

186.0 km186.0 km = 4 sig dig= 4 sig dig

1.86 x 101.86 x 1055 m m = 3 sig dig= 3 sig dig

1.86000 x 101.86000 x 1055 m m = 6 sig dig= 6 sig dig

0.186 Mm0.186 Mm = 3 sig dig= 3 sig dig

0.000186 Gm0.000186 Gm = 3 sig dig= 3 sig dig

Rules to Determine # of Sig DigRules to Determine # of Sig Dig

1. Nonzeros are always significant.1. Nonzeros are always significant.

2. All final zeros after the decimal point are 2. All final zeros after the decimal point are significant.significant.

3. Zeros between two other sig dig are 3. Zeros between two other sig dig are always significant.always significant.

4. Zeros used solely as placeholders are 4. Zeros used solely as placeholders are not significant.not significant.

EXAMPLES # Sig DigEXAMPLES # Sig Dig

450 026450 026 = = 660.123 =0.123 =33100 258 =100 258 =660.000 009 =0.000 009 =110.000 090 =0.000 090 =22

Addition & SubtractionAddition & Subtraction

Perform the operation, then round off the least Perform the operation, then round off the least precise value involved.precise value involved. 64.032464.0324 9.6419.641 + + 129 458.1129 458.1 = 129 531.7734= 129 531.7734

129 458.1 is the least precise value 129 458.1 is the least precise value round off to 129 531.8 ,one digit past the round off to 129 531.8 ,one digit past the decimal point.decimal point.

Multiplication & DivisionMultiplication & Division

Perform the calculation, round the product Perform the calculation, round the product or quotient to the factor with the least or quotient to the factor with the least significant digit.significant digit.

4.631cm x 7.2cm = 33.3432cm4.631cm x 7.2cm = 33.3432cm22

33cm33cm22

3.67 x 1.9 = 6.9733.67 x 1.9 = 6.973

7.0 7.0

More ExamplesMore Examples

29.4m 29.4m ÷ 2.431s = 12. 09378856m/s÷ 2.431s = 12. 09378856m/s

12.1m/s12.1m/s

143 004 + 16.235 + 7.04 + 98.0357 + 0.1 143 004 + 16.235 + 7.04 + 98.0357 + 0.1 = 143 135. 4107= 143 135. 4107

143 135143 135

Examples Cont.Examples Cont.

142.65 - 73.976 = 68.674142.65 - 73.976 = 68.674

68.6768.67

15.003 x 29.745 x 0.62 x 145 15.003 x 29.745 x 0.62 x 145 = 40 119.15473= 40 119.15473

40 000 (ambiguous)40 000 (ambiguous)

More correct 4.0 x 10More correct 4.0 x 1044

Examples Cont.Examples Cont.

62 579 62 579 ÷ 0.37 = 169 132.4324÷ 0.37 = 169 132.4324

170 000170 000

More correct 1.7 x 10More correct 1.7 x 1055

Rounding with “5”Rounding with “5”

Rounding off when “5” is the last digitRounding off when “5” is the last digit

142.55 & 142.65142.55 & 142.65

142.55 = 142.6142.55 = 142.6

142.65 = 142.6142.65 = 142.6

When an even number precedes the 5, When an even number precedes the 5, round “down”, when odd number precedes round “down”, when odd number precedes the 5 round “up”the 5 round “up”

Section 2.3 Visualizing DataSection 2.3 Visualizing Data

Graphing DataGraphing DataIndependent Variable =Independent Variable =The variable that is changed or manipulated The variable that is changed or manipulated

directly by the experimenter.directly by the experimenter.Dependent Variable =Dependent Variable =A result of a A result of a ΔΔ the independent variable, the independent variable,

AKA, the responding variable. The value AKA, the responding variable. The value of the dependent variable “DEPENDS” on of the dependent variable “DEPENDS” on the the ΔΔ the independent variable. the independent variable.

Rules for Plotting Line GraphsRules for Plotting Line Graphs

1. Identify the IV and DV. The IV is plotted 1. Identify the IV and DV. The IV is plotted on the horizontal, x-axis, and the DV is on the horizontal, x-axis, and the DV is plotted on the vertical, y-axis.plotted on the vertical, y-axis.

2. Determine the range of the IV to be 2. Determine the range of the IV to be plotted.plotted.

3. Decide where the graph begins, the 3. Decide where the graph begins, the origin (0,0) is NOT ALWAYS the starting origin (0,0) is NOT ALWAYS the starting point.point.

4. Spread the data out as much as 4. Spread the data out as much as possible. Let each division on the graph possible. Let each division on the graph paper stand for a convenient unit.paper stand for a convenient unit.

5. Number and 5. Number and LABELLABEL the horizontal axis. the horizontal axis.

6. Repeat steps 2-5 for the DV.6. Repeat steps 2-5 for the DV.

7. Plot the data points on the graph.7. Plot the data points on the graph.

8. Draw the “BEST FIT 8. Draw the “BEST FIT LINE”, LINE”, straightstraight or a or a smooth smooth curvecurve, that passes through , that passes through as many data points as as many data points as possible. Keep in mind the possible. Keep in mind the best fit line best fit line may notmay not pass pass through any points. Do not through any points. Do not draw a series of straight draw a series of straight lines that simply “connect lines that simply “connect the dots”.the dots”.

9. Give the graph a title that 9. Give the graph a title that CLEARLYCLEARLY tells what the tells what the graph represents.graph represents.

Linear RelationshipsLinear Relationships

A graph where a straight line can be A graph where a straight line can be drawn through drawn through ALLALL the points. the points.

The two variables are directly proportional; The two variables are directly proportional; as as xx increases, increases, yy also increases by the also increases by the same % and as same % and as xx decreases, decreases, yy decreases decreases by the same %.by the same %.

Example of Linear GraphExample of Linear Graph

Slope = Linear GraphSlope = Linear Graph

y y = m= mx x + b+ b

m = slope, the ratio of vertical m = slope, the ratio of vertical ΔΔ to to horizontal horizontal ΔΔ

b = b = yy-intercept, the point at which the line -intercept, the point at which the line crosses the crosses the yy-axis and it is the -axis and it is the yy value value when when xx = 0. = 0.

riserise ΔΔ yy yyff - - yyii

m = run = m = run = ΔΔxx = = xxff – – xxii

Slope can be negative, Slope can be negative, yy gets smaller as gets smaller as xx gets bigger. gets bigger.

ExampleExample

TRANSPARENCY EXAMPLESTRANSPARENCY EXAMPLES

Calculate slopeCalculate slope

Which slope is greaterWhich slope is greater

Nonlinear RelationshipsNonlinear Relationships

A graph that produces a smooth curved A graph that produces a smooth curved line.line.

Sometimes a parabola where the two Sometimes a parabola where the two variables are related by a quadratic variables are related by a quadratic relationship:relationship:

yy = a = axx22 + b + bxx + c + c

Also expressed as aAlso expressed as axx22 + b + bxx + c = 0 + c = 0

One variable depends One variable depends on the square of the on the square of the other.other.

Sometimes produces a Sometimes produces a graph that is inversely graph that is inversely proportional, the graph proportional, the graph is a hyperbola. is a hyperbola.

Inverse relationship Inverse relationship y = a/x or xy = a y = a/x or xy = a

As one variable As one variable increases the other increases the other variable decreases.variable decreases.