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Prerequisite Skills,Properties of Rea~NumbersFor any real numbers a, b, and c, the following properties holdtrue.

Closure Property a + band ab are real numbers. 2 and 4 are real, so 2 + 4or 6 is real and 2 . 4 or 8is real.

Commutative Property a + b = b + a and ab = ba 4 + 7 = 7 + 4;5· 8 = 8· 5

Associative Property a + (b + c) = (a + b) + c 4 + (7 +9) = (4 + 7) + 9;and a(bc) = (ab)c 3(6 . 5) = (3 . 6)5

Distributive Property a(b + e) = ab + ae 12(x + 4) = 12x + 48

Additive Identity a+O=a 3x+ 0 = 3xMultiplicative Identity a·1 = a 7Y'1 =7yAdditive Inverse For each real number a, there 3+(-3)=0

is a unique real number -a

0 such that a + (-a) = O.~. Multiplicative Inverse For each real number a -=I=-0, 17·- = 1

there is a unique real number..!..7

asuch that a . ..!..= 1.

a

Multiplication Property a· 0 = 0 8x· 0 = 0Of ZeroReflexive Property a=a 9=9Of EqualitySymmetric Property If a = b, then b = a. If 3 = x, then x = 3.Of EqualityTransitive Property If a = band b = c, then a = e. If x = 2 and y = 2,Of Equality then x = y:Substitution Property If a = b, then a and b may be If x = 2y and z = 3x,Of Equality substituted for one another in then z = 3(2.0.

any expression involving a or b.Trichotomy Property For any two real numbers x < 6, or x = 6, or x > 6

a and b, exactly one of thefollowing is true: a < b, a = b,or a> b.

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© Gfencoe/McGraw-HilI 3 Advanced Mathematical Concepts

NAIVIE _ DATE -.,- PERIOD _ II

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t.-:r- -Study Guide,Modeling Real-World Data with Linear FunctionsWhen real-world data are collected,the data graphed usually do notform a straight line. However, the graph may approximate a linear - {

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fOS,hve, ixtl6 ~q:p-hvext~.tt noneCorrelation Coefficient: ( r) where r is betweeill< r $1, is a measure that showshow well the. data points are modeled by a linear equation.

• If r is negative, the data points have a negative correlation• If r is positive; the data points have a positive correlation

\. If r is 0, the data has no correlation

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Calculator instructions-Regression Line (Line of Best Fit):2nd 0: (catalog) diagnostic on (this will turn on your r, correlation coefficient)EnterStatEdit ~ and L2 (this puts data in List 1 (x's) and List 2 (y's»2nd quit2nd y = (stat plot) this will show you what your data looks likeEnterPlot 1 (on) should be highlightedType+:~::~ Highlight _

EnterZoom 9GraphIs it a line? If it is, get an equationStatCalc4: linReg( ax+b)Enter

t XList: 4YList: t;Store RegEQ: VarsY-Vars1:Function (enter)Yr

EnterCalculate (enter)

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Prert:quisite Skills Practice

Complete each statement using the given' property.

1. 5(4' 9) = ,Associative PropertyLinear Regression

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2. m + 0 = ,Additive Identity

Data: Is there a relationship between MathSAT scores and the number of hours spentstudying for the test? A study was conductedinvolving 20 students as they prepared for andtook the Math section of the SAT Examination.

Task:

a.) Determine a linear regression modelequation to represent this data.

b.} Graph this new equation. .Be sure to label the axes andindicate a scale.

10 640

~~~~~~&~S.~~~~~~4 390

3. 6x = , Commutative Property

4. If 8 = Sn, then , Symmetric Property9 58010 650 5. 4(x _ 6) = , Distributive Property14 7304 410 6. 8 + y = , Commutative Property7 53012 600 7.Ift '" 7, then either or , Trichotomy Property22 7901 350

Name the property that justifies each statement8. 3 is a real number. If x is a real number, then 3J;is a real number.

3 4008 59011 6405 450 9. If:t = 7, then 6:r "= 6(7).6 52010 690 10.9n' 1 = 9"II 69016 770 11. x = 8 and 8 = 2y, thus x = 2y13 700

12. Given that-x ., 12, it must be that x < 12 or x > 12.IJ 730

13. 4 .-72 = 4(70 + 2) = 4 . 70 + 4 . 2

14.17·0=0

15. (5" + 7) + 4 = 5" + (7 + 4)

c.) Is the new equation a "good fit" to represent this data?Explain how you made your decision. -----

d.) Interpolate data: If a student studied for 15 hours, based UPOD tbis study,what would be the expected Math SAT score? --'- _

e.) Interpolate data: If a student obtained a Math SAT score of 720, based uponthis study, how many hours did the student most likely spend studying?

f.) Extrapolate data: If a student spent 100 hours studying, what would be theexpected Math SAT score?_~ymm Ili:d !biis;.=-a=-JIt::ll---;n:-:"'""=='"--••••- ••-"'-mrll--;-Ay-=--_-----