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## Chapter 1 Real Numbers and Algebra

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**Chapter 1**Real Numbers and Algebra**1.1 Describing Data with Set of numbers**• Natural Numbers are counting numbers and can be expressed as N = { 1, 2, 3, 4, 5, 6, …. } Set braces { }, are used to enclose the elements of a set. • A whole numbers is a set of numbers, is given by W = { 0, 1, 2, 3, 4, 5, ……}**…Continued**• The set of integers include both natural and the whole numbers and is given by I = { …, -3, -2, -1, 0, 1, 2, 3, ….} • A rational number is any number can be written as the ratio of two integers p/q, where q = 0. Rational numbers can be written as fractions and include all integers. Some examples of rational numbers are 8/1, 2/3, -3/5, -7/2, 22/7, 1.2, and 0.**…continued**• Rational numbers may be expressed in decimal form that either repeats or terminates. The fraction 1/3 may be expressed as 0.3, a repeating decimal, and the fraction ¼ may be expressed as 0.25, a terminating decimal. The overbar indicates that0.3 = 0.3333333…. • Some real numbers cannot be expressed by fractions. They are called irrational numbers. 2, 15, and are examples of irrational numbers.**Identity Properties**For any real number a, • a + 0 = 0 + a = a, 0 is called the additive identityand • a . 1 = 1 . a = a, The number 1 is called the multiplicative identity. • Commutative Properties For any real numbers a and b, a + b = b + a(Commutative Properties of addition) a.b = b.a(Commutative Properties of multiplication)**…Continued**Associative Properties For any real numbers a, b, c, (a + b) + c = a + (b + c)(Associative Properties of addition) (a.b) . c = a . (b . c)(Associative Properties for multiplication) Distributive Properties For any real numbers a, b, c, a(b + c) = ab + acand a(b- c) = ab - ac**1.2 Operation on Real Numbers**The Real Number Line -3 -2 -1 0 1 2 3 Origin -2 2 -2 = 2Absolute value cannot be negative 2 = 2 -3 -2 -1 0 1 2 3 Origin**…Continued**If a real number a is located to the left of a real number b on the number line, we say that a is less than b and writea<b. Similarly, if a real number a is located to the right of a real number b, we say that a is greater than b and writea>b. Absolute value of a real number a, written a , is equal to its distance from the origin on the number line. Distance may be either positive number or zero, but it cannot be a negative number.**Arithmetic Operations**Addition of Real Numbers To add two numbers that are either both positive or both negative, add their absolute values. Their sum has the same sign as the two numbers. Subtraction of real numbers For any real numbersa and b, a-b = a + (-b). Multiplication of Real Numbers The product of two numbers with like signs is positive. The product of two numbers with unlike signs is negative. Division of Real Numbers For real numbers aand b, with b = 0, a/b = a . 1/b That is, to divide a by b, multiply a by the reciprocal of b.**1.3 Bases and Positive Exponents**Squared 4 Cubed 4 4 4 44 4 . 4 = 42 4 . 4. 4 = 43 Exponent Base**1.3 Integer Exponents**Let a be a nonzero real number and n be a positive integer. Then an = a. a. a. a……a (n factors of a ) a0 = 1, and a –n = 1/an a -n b m b -m = a n a -n b n b = a**… cont**• The Product Rule For any non zero number a and integers m and n, am . an = a m+n • The Quotient Rule For any nonzero number a and integers m and n, am = a m – n a n**Raising Products To Powers**For any real numbers a and b and integer n, (ab) n = a n b n • Raising Powers to Powers For any real number a and integers m and n, (am)n = a mn Raising Quotients to Powers For nonzero numbers a and b and any integer n. a n = an b bn**…Continued**A positive number a is in scientific notation when a is written as b x 10n, where 1 < b < 10 and n is an integer. • Scientific Notation Example : 52,600 = 5.26 x 104 and 0.0068 = 6.8 x 10 -3**1.4 Variables, Equations , and Formulas**• A variableis a symbol, such as x, y, t, used to represent any unknown number or quantity. • An algebraic expression consists of numbers, variables, arithmetic symbols, paranthesis, brackets, square roots. Example 6, x + 2, 4(t – 1)+ 1, X + 1**…cont**• An equationis a statement that says two mathematical expressions are equal. Examples of equation 3 + 6 = 9, x + 1 = 4, d = 30t, and x + y = 20 • A formula is an equation that can be used to calculate one quantity by using a known value of another quantity. The formula y = x/3computes the no. of yards in x feet. If x= 15, then y=15/3= 5.**Square roots**The number b is a square root of a number a if b2 = a. Example - One square root of 9 is 3 because 32 = 9. The other square root of 9 is –3 because (-3)2 = 9. We use the symbol to 9 denote the positive or principal square root of 9. That is, 9 = +3. The following are examples of how to evaluate the square root symbol. A calculator is sometimes needed to approximate square roots, 4 = + 2 - The symbol ‘ + ‘ is read ‘plus or minus’. Note that 2 - represents the numbers 2 or –2.**Cube roots**The number b is a cube root of a number a if b3 = a The cube root of 8 is 2 because 23 = 8, which may be written as 3 8 = 2. Similarly 3 –27 = -3 because (- 3)3 = - 27. Each real number has exactly one cube root.**1.5 Introduction to graphing**Relations is a set of Ordered pairs. If we denote the ordered pairs in a relation (x,y), then the set of all x-values is called the domainof the relation and the set of all y values is called the range.**Example 1.**Find the domain and range for the relation given by S = {( -1, 5), (0,1), (2, 4), (4,2), (5,1)}Solution The domain D is determined by the first element in each ordered pair, or D ={-1, 0, 2, 4,5} The rangeR is determined by the second element in each ordered pair, or R = {1,2,4,5}**The Cartesian Coordinate System**Quadrant II y Quadrant I y (1, 2) 2 1 0 -1 -2 Origin 2 1 -1 -2 x x -2 -1 1 2 -2 -1 1 2 Quadrant III Quadrant IV The xy – plane Plotting a point**Scatterplots and Line Graphs**If distinct points are plotted in the xy- plane, the resulting graph is called a scatterplot. Y 7 6 5 4 3 2 1 (4, 6) (3, 5) (5, 4) (2, 4) (6, 3) (1, 2) X 0 1 2 3 4 5 6 7**Ex 9 page 44**Make a table for y = x 2/ 9, starting at x = 10 and incrementing by 10 and compare The table for example 4 ( pg 41) Go to Y= and enter x 2/9 Go to 2ndthen table set and enter Go to 2nd then table**Viewing Rectangle ( Page 57 )**Ymax }Ysc1 Xmax Xmin Xsc1 Ymin [ -2, 3, 0.5] by [-100, 200, 50]**Making a scatterplot with a graphing calculator**Plot the points (-2, -2), (-1, 3), (1, 2) and (2, -3) in [ -4, 4, 1] by [-4, 4, 1] (Example 10, page 58) Go to 2ndthen stat plot Go to Stat Edit then enter points Scatter plot [ -4, 4, 1] by [-4, 4, 1]**Example 11Cordless Phone Sales**Enter line graph Go to Stat edit and enter data Hit graph Enter datas in window [1985, 2002, 5] by [0, 40, 10]**Chapter 2**Linear Functions and Models**Ch 2.1 Functions and Their Representations**A function is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value. Input x Output y Function f (x, y) Input Output**…continued**y is a function of x because the output y is determined by and depends on the input x. As a result, y is called the dependent variable and x is the independent variable To emphasize that y is a function of x, we use the notation y = f(x) and is called a function notation. A function f forms a relation between inputs x and outputs y that can be represented verbally (Words) , numerically (Table of values) , symbolically (Formula), and graphically (Graph).**Representation of Function**y Table of Values Graph 20 16 12 8 4 y= 3x x 0 4 8 12 16 20 24 Numerically Graphically**Diagrammatic Representation**Function Not a function x y x y 3 6 9 (1, 3), (2, 6), (3, 9) 1 2 3 4 5 6 1 2 1 2 3 4 5 (1, 4), (2, 5), (2, 6) (1,4), (2, 4), (3, 5) Example 1, pg - 77**Domain and RangeGraphically**The domainof f is the set of all x- values, and the range of f is the set of all y-values y 3 2 1 Range R includes all y – values satisfying 0 < y < 3 x Domain D includes all x values Satisfying –3 < x < 3 Range -3 -2 -1 0 1 2 3 Domain Pg 79 Ex-5**Vertical Line Test**If each vertical line intersects the graph at most once, then it is a graph of a function 5 4 3 2 1 -1 -2 -3 -4 -5 Not a function (-1, 1) -4 -3 -2 -1 0 1 2 3 4 (-1, -1)**…Continued**Not a function 4 3 2 1 -1 -2 -3 (-1, 1) -3 -2 -1 0 1 2 3 (1, -1) Example 9 page - 81**Using Technology**Graph of y = 2x - 1 Hit Y and enter 2x - 1 Hit 2nd and hit table and enter data [ - 10, 10, 1] by [ - 10, 10, 1]**2.2 Linear Function**A function f represented by f(x) = ax + b, where a and b are constants, is a linear function. 100 90 80 70 60 100 90 80 70 60 f(x) = 2x + 80 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Scatter Plot A Linear Function Ex- 1, 2, 3, 4, 5 Pg 91**Modeling data with Linear Functions**1500 1250 1000 750 500 250 0 Example 7 Cost (dollars) • 8 12 16 20 x • Credits Symbolic Representation f(x) = 80x + 50 Numerical representation 4 8 12 16 $ 370 $ 690 $1010 $1330**Using a graphing calculator**Example 5 Give a numerical and graphical representation f(x) = ½ x - 2 Numerical representation –Y1 = .5x – 2 starting x = -3 Graphical representation – [ -10, 10, 1] by [-10, 10, 1]**2.3 The Slope of a line**Y 8 7 6 5 4 3 2 1 Cost (dollars) Rise = 3 Slope = Rise = 3 Run 2 Run = 2 1 2 3 4 5 6 x Gasoline (gallons ) Cost of Gasoline Every 2 gallons purchased the cost increases by $3**2.3 Slope**rise y2 - y1 m = run = x2 - x1 The Slope m of the line passing through the points (x1 y1 ) and (x2, y2) is m= y2 –y1/x2 –x1 Where x1 = x2. That is, slope equals rise over run. y2 (x2, y2) y2 –y1 y1 (x1, y1) x2 –x1 Rise Run Ex- 1, 2 pg 104**4**3 2 1 0 -1 -2 4 3 2 1 -1 -2 -3 2 -1 m = - ½ < 0 m = 2 > 0 2 -4 -2 1 2 3 4 -4 -2 1 2 Positive slope Negative slope m = 0 m is undefined Zero slope Undefined slope**(0, 4)Example 2 - Sketch a line passing through**the point (0, 4) and having slope - 2/3y - valuesdecrease 2 units each times x- values increase by 3(0 + 3, 4 – 2)= (3, 2) • 4 • 3 • 2 • 1 • 1 • 2 • 3 • 4 ( 0, 4) Rise = -2 ( 3, 2) - 4 - 3 - 2 1 0 1 2 3 4**Slope-Intercept Form**The line with slope m and y = intercept b is given by y= mx + b The slope- Intercept form of a line Example – 3, and 4, 5, 6, 7 pg - 106**Example - 3**3 2 1 -1 -2 -3 Y = ½ x Y = ½ x + 2 -3 -2 -1 1 2 Y = ½ x - 2**Analyzing Growth in Walmart**Year 1997 1999 2002 2007 Employees 0.7 1.1 1.4 2.2 Example 9 3.0 2.5 2.0 1.5 1.0 0.5 Employees (millions) m3 m2 m1 Years 0 1999 2003 2007 m1= 1.1 – 0.7 = 0.2m2 = 1.4 - 1.1 = 0.1 and 1999 – 1997 2002 – 1999 m3 = 2.2 - 1.4 = 0.16 2007 - 2002 Average increase rate**2.4 Point- slope form**The line with slope m passing through the point (x1 , y1 ) is given by y = m ( x - x1)+ y1 Or equivalently, y – y1 = m (x –x1) The point- slope form of a line (x, y) y – y1 (x1, y1) x – x1 m = y – y1 / x – x1 Ex 1, 2,3 pg 117**Horizontal and Vertical Lines**Equation of vertical line y y x = h x x b h y= b Equation of Horizontal Line**…Continued**Parallel Lines Two lines with the same slope are parallel. m1 = m2 Perpendicular Lines Two lines with nonzero slopes m1 and m2are perpendicular if m1 m2= -1 Examples – 7, 8 Page - 123